Vim, 57 bytes

:%s/a/ &

:%norm +hkyiwjP

:g/d/norm diw-@"yl+P

:%s/ //g

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JavaScript (ES6),  66 62  61 bytes

a=>a.map(p=s=>a=a.slice([,x,y]=/(d*)(.*)/.exec(s),p=x||p)+y)

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Commented

a =>                  // a = input, re-used to store the previous word

  a.map(p =           // initialize p to a non-numeric value

  s =>                // for each string s in a:

    a =               //   update a:

      a.slice(        //     extract the correct prefix from the previous word:

        [, x, y] =    //       load into x and y:

          /(d*)(.*)/ //         the result of a regular expression which splits the new

          .exec(s),   //         entry into x = leading digits and y = trailing letters

                      //       this array is interpreted as 0 by slice()

        p = x || p    //       update p to x if x is not an empty string; otherwise leave

                      //       it unchanged; use this as the 2nd parameter of slice()

      )               //     end of slice()

      + y             //     append the new suffix

  )                   // end of map()

Perl 6, 50 48 bytes

-2 bytes thanks to nwellnhof

{my$l;.map:{$!=S[d*]=substr $!,0,$l [R||]=~$/}}

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A port of Arnauld's solution. Man, that R|| trick was a rollercoaster from 'I think this could be possible', to 'nah, it's impossible', to 'kinda maybe possible' and finally 'aha!'

Explanation:

{my$l;.map:{$!=S[d*]=substr $!,0,$l [R||]=~$/}}

{                                              }  # Anonymous code block

 my$l;    # Declare the variable $l, which is used for the previous number

      .map:{                                  }  # Map the input list to

            $!=              # $! is used to save the previous word

               S[d*]=       # Substitute the number for

                      substr $!,0    # A substring of the previous word

                                 ,              # With the length of 

                                           ~$0     # The num if it exists

                                  $l [R||]=        # Otherwise the previous num

The $l [R||]=~$/ part roughly translates to $l= ~$/||+$l but... it has the same amount of bytes :(. Originally, it saved bytes using an anonymous variable so the my$l was gone but that doesn't work since the scope is now the substitution, not the map codeblock. Oh well. Anyways, R is the reverse metaoperator, so it reverses the arguments of ||, so the $l variable ends up being assigned the new number (~$/) if it exists, otherwise itself again.

It could be 47 bytes if Perl 6 didn't throw a kinda redundant compiler error for =~.

Ruby, 49 45 43 bytes

$0=$_=$0[/.{0#{p=$_[/d+/]||p}}/]+$_[/D+/]

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Explanation

$0=                                         #Previous word, assign the value of

   $_=                                      #Current word, assign the value of

      $0[/.{0#{              }}/]           #Starting substring of $0 of length p which is

               p=$_[/d+/]||p               #defined as a number in the start of $_ if any 

                                 +$_[/D+/] #Plus any remaining non-digits in $_

Python 3.6+, 172 195 156 123 122 121 104 bytes

import re

def f(l,n=0,w=""):

 for s in l:t=re.match("d*",s)[0];n=int(t or n);w=w[:n]+s[len(t):];yield w

• -12 bytes thanks to Bubbler
  


• -5 bytes thanks to Dennis
  

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Explanation

I caved, and used Regular Expressions. This saved at least 17 bytes. :

t=re.match("d*",s)[0]

When the string doesn't begin with a digit at all, the length of this string will be 0. This means that:

n=int(t or n)

will be n if t is empty, and int(t) otherwise.

w=w[:n]+s[len(t):]

removes the number that the regular expression found from s (if there's no number found, it'll remove 0 characters, leaving s untruncated) and replaces all but the first n characters of the previous word with the current word fragment; and:

yield w

outputs the current word.

C, 65 57 bytes

n;f(){char c[99];while(scanf("%d",&n),gets(c+n))puts(c);}

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Explanation:

n;                     /* n is implicitly int, and initialized to zero. */



f() {                  /* the unpacking function. */



    char c[99];        /* we need a buffer to read into, for the longest line in

                          the full dictionary we need 12 + 1 bytes. */



    while(             /* loop while there is input left. */



        scanf("%d",&n) /* Read into n, if the read fails because this line

                          doesn't have a number n's value does not change.

                          scanf's return value is ignored. */



        ,              /* chain expressions with the comma operator. The loop

                          condition is on the right side of the comma. */



        gets(c+n))     /* we read into c starting from cₙ. c₀, c₁.. up to cₙ is

                          the shared prefix of the word we are reading and the

                          previous word. When gets is successful it returns c+n

                          else it will return NULL. When the loop condition is

                          NULL the loop exits. */



        puts(c);}      /* print the unpacked word. */

Haskell, 82 81 bytes

tail.map concat.scanl p["",""]

p[n,l]a|[(i,r)]<-reads a=[take i$n++l,r]|1<2=[n,a]

Takes and returns a list of strings.

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        scanl p["",""]        -- fold function 'p' into the input list starting with

                              -- a list of two empty strings and collect the

                              -- intermediate results in a list

  p [n,l] a                   -- 1st string of the list 'n' is the part taken form the last word

                              -- 2nd string of the list 'l' is the part from the current line

                              -- 'a' is the code from the next line

     |[(i,r)]<-reads a        -- if 'a' can be parsed as an integer 'i' and a string 'r'

       =[take i$n++l,r]       -- go on with the first 'i' chars from the last line (-> 'n' and 'l' concatenated) and the new ending 'r'

     |1<2                     -- if parsing is not possible

       =[n,a]                 -- go on with the previous beginning of the word 'n' and the new end 'a'

                              -- e.g. [         "aa",     "2h",      "3ed",       "ing"       ] 

                              -- ->   [["",""],["","aa"],["aa","h"],["aah","ed"],["aah","ing"]]

  map concat                  -- concatenate each sublist

tail                          -- drop first element. 'scanl' saves the initial value in the list of intermediate results. 

Edit: -1 byte thanks to @Nitrodon.

Japt, 19 18 17 bytes

Initially inspired by Arnauld's JS solution.

;£=¯V=XkB ªV +XoB

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                      :Implicit input of string array U

 £                    :Map each X

   ¯                  :  Slice U to index

      Xk              :    Remove from X

;       B             :     The lowercase alphabet (leaving only the digits or an empty string, which is falsey)

          ªV          :    Logical OR with V (initially 0)

    V=                :    Assign the result to V for the next iteration

             +        :  Append

              Xo      :  Remove everything from X, except

;               B     :   The lowercase alphabet

  =                   :  Reassign the resulting string to U for the next iteration

brainfuck, 201 bytes

,[[[-<+>>>+<<]>-[---<+>]<[[-<]>>]<[-]>>[<<,>>>[-[-<++++++++++>]]++++<[->+<]-[----->-<]<]<]>>>[[>>]+[-<<]>>[[>>]+[<<]>>-]]+[>>]<[-]<[<<]>[->[>>]<+<[<<]>]>[>.>]+[>[-]<,.[->+>+<<]>>----------]<[<<]>-<<<,]

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Requires a trailing newline at the end of the input. A version without this requirement is 6 bytes longer:

brainfuck, 207 bytes

,[[[-<+>>>+<<]>-[---<+>]<[[-<]>>]<[-]>>[<<,>>>[-[-<++++++++++>]]++++<[->+<]-[----->-<]<]<]>>>[[>>]+[-<<]>>[[>>]+[<<]>>-]]+[>>]<[-]<[<<]>[->[>>]<+<[<<]>]>[>.>]+[>[-]<,[->+>+<<]>>[----------<.<]>>]<[<<]>-<<<,]

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Both versions assume all numbers are strictly less than 255.

Explanation

The tape is laid out as follows:

tempinputcopy 85 0 inputcopy number 1 a 1 a 1 r 1 d 0 w 0 o 0 l 0 f 0 ...

The "number" cell is equal to 0 if no digits are input, and n+1 if the number n is input. Input is taken at the cell marked "85".

,[                     take input and start main loop

 [                     start number input loop

  [-<+>>>+<<]          copy input to tempinputcopy and inputcopy

  >-[---<+>]           put the number 85 in the cell where input was taken

  <[[-<]>>]            test whether input is less than 85; ending position depends on result of comparison

                       (note that digits are 48 through 57 while letters are 97 through 122)

  <[-]>                clean up by zeroing out the cell that didn't already become zero

  >[                   if input was a digit:

   <<,>>               get next input character

   >[-[-<++++++++++>]] multiply current value by 10 and add to current input

   ++++                set number cell to 4 (as part of subtracting 47)

   <[->+<]             add input plus 10*number back to number cell

   -[----->-<]         subtract 51

  <]                   move to cell we would be at if input were a letter

 <]                    move to input cell; this is occupied iff input was a digit



                       part 2: update/output word



 >>>                   move to number cell

 [                     if occupied (number was input):

  [>>]+[-<<]>>         remove existing marker 1s and decrement number cell to true value

  [[>>]+[<<]>>-]       create the correct amount of marker 1s

 ]

 +[>>]<[-]             zero out cell containing next letter from previous word

 <[<<]>                return to inputcopy

 [->[>>]<+<[<<]>]      move input copy to next letter cell

 >[>.>]                output word so far

 +[                    do until newline is read:

  >[-]<                zero out letter cell

  ,.                   input and output next letter or newline

  [->+>+<<]            copy to letter cell and following cell

  >>----------         subtract 10 to compare to newline

 ]

 <[<<]>-               zero out number cell (which was 1 to make copy loop shorter)

 <<<,                  return to input cell and take input

]                      repeat until end of input

Jelly, 16 bytes

⁹fØDVo©®⁸ḣ;ḟØDµ

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How it works

⁹fØDVo©®⁸ḣ;ḟØDµ  Main link. Argument: A (array of strings)



              µ  Cumulatively reduce A by the link to the left.

⁹                     Yield the right argument.

  ØD                  Yield "0123456789".

 f                    Filter; keep only digits.

    V                 Eval the result. An empty string yields 0.

     o©               Perform logical OR and copy the result to the register.

       ®              Yield the value in the register (initially 0).

        ⁸ḣ            Head; keep that many character of the left argument.

          ;           Concatenate the result and the right argument.

            ØD        Yield "0123456789".

           ḟ          Filterfalse; keep only non-digits.

Python 2, 118 bytes

import re

n=0

l=input()

o=l.pop(0)

print o

for i in l:(N,x),=re.findall('(d*)(.+)',i);n=int(N or n);o=o[:n]+x;print o

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Retina 0.8.2, 69 bytes

+`((d+).*¶)(D)

$1$2$3

d+

$*

+m`^((.)*(.).*¶(?<-2>.)*)(?(2)$)1

$1$3

Try it online! Link includes harder test cases. Explanation:

+`((d+).*¶)(D)

$1$2$3

For all lines that begin with letters, copy the number from the previous line, looping until all lines begin with a number.

d+

$*

Convert the number to unary.

+m`^((.)*(.).*¶(?<-2>.)*)(?(2)$)1

$1$3

Use balancing groups to replace all 1s with the corresponding letter from the previous line. (This turns out to be slightly golfier than replacing all runs of 1s.)

Red, 143 bytes

func[b][a: charset[#"a"-#"z"]u: b/1 n: 0 foreach c b[parse c[copy m to a

p: copy s to end(if p<> c[n: do m]print u: rejoin[copy/part u n s])]]]

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Java (JDK), 150 bytes

a->{String p="",s;for(int n=0,i=0;i<a.length;a[i]=p=p.substring(0,n=s.length<1?n:new Short(s[0]))+a[i++].replaceAll("\d",""))s=a[i].split("\D+");}

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Groovy, 74 bytes

{w="";d=0;it.replaceAll(/(d*)(.+)/){d=(it[1]?:d)as int;w=w[0..<d]+it[2]}}

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Explanation:

{                                                                        }  Closure, sole argument = it

 w="";d=0;                                                                  Initialize variables

          it.replaceAll(/(d*)(.+)/){                                   }   Replace every line (since this matches every line) and implicitly return. Loop variable is again it

                                     d=(it[1]?:d)as int;                    If a number is matched, set d to the number as an integer, else keep the value

                                                        w=w[0..<d]+it[2]    Set w to the first d characters of w, plus the matched string

Jelly, 27 bytes

f€ȯ@V,ɗḟ€ɗØDZẎḊṖḣ2/Ż;"f€Øa

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Perl 5 -p, 45 41 bytes

s:d*:substr($p,0,$l=$&+$l*/^D/):e;$p=$_

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Explanation:

s:d*:substr($p,0,$l=$&+$l*/^D/):e;$p=$_ Full program, implicit input

s:   :                           :e;      Replace

  d*                                       Any number of digits

      substr($p,0,              )           By a prefix of $p (previous result or "")

                  $l=  +                      With a length (assigned to $l) of the sum

                     $&                         of the matched digits

                          *                     and the product

                        $l                        of $l (previous length or 0)

                           /^D/                  and whether there is no number in the beginning (1 or 0)

                                                (product is $l if no number)

                                    $p=$_ Assign output to $p

                                          Implicit output

Groovy, 103 99 bytes

{w=it[0];d=0;it.collect{m=it=~/(d+)(.+)/;i=m.find()?{d=m[0][1] as int;m[0][2]}():it;w=w[0..<d]+i}}

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05AB1E, 20 19 17 bytes

õUvyþDõÊi£U}Xyá«=

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Explanation:

õ                  # Push an empty string ""

 U                 # Pop and store it in variable `X`

v                  # Loop `y` over the (implicit) input-list

 yþ                #  Push `y`, and leave only the digits (let's call it `n`)

   DõÊi  }         #  If it's NOT equal to an empty string "":

       £           #   Pop and push the first `n` characters of the string

        U          #   Pop and store it in variable `X`

          X        #  Push variable `X`

           yá      #  Push `y`, and leave only the letters

             «     #  Merge them together

              =    #  Print it (without popping)

Common Lisp, 181 bytes

(do(w(p 0))((not(setf g(read-line t()))))(multiple-value-bind(a b)(parse-integer g :junk-allowed t)(setf p(or a p)w(concatenate'string(subseq w 0 p)(subseq g b)))(format t"~a~%"w)))

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Ungolfed:

(do (w (p 0))   ; w previous word, p previous integer prefix (initialized to 0)

    ((not (setf g (read-line t ()))))   ; read a line into new variable g

                                        ; and if null terminate: 

  (multiple-value-bind (a b)            ; let a, b the current integer prefix

      (parse-integer g :junk-allowed t) ; and the position after the prefix

    (setf p (or a p)                    ; set p to a (if nil (no numeric prefix) to 0)

          w (concatenate 'string        ; set w to the concatenation of prefix

             (subseq w 0 p)             ; characters from the previous word 

             (subseq g b)))             ; and the rest of the current line

    (format t"~a~%"w)))                 ; print the current word

As usual, the long identifers of Common Lisp make it non particularly suitable for PPCG.

Python 2, 101 100 99 bytes

import re

s=n='0'

for t in input():(m,w),=re.findall('(d*)(.+)',t);n=m or n;s=s[:int(n)]+w;print s

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C# (Visual C# Interactive Compiler), 134 bytes

a=>{int l=0,m,n;var p="";return a.Select(s=>{for(m=n=0;s[m]<58;n=n*10+s[m++]-48);return p=p.Substring(0,l=m>0?n:l)+s.Substring(m);});}

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-9 bytes thanks to @ASCIIOnly!

Less golfed...

// a is an input list of strings

a=>{

  // l: last prefix length

  // m: current number of digits

  // n: current prefix length

  int l=0,m,n;

  // previous word

  var p="";

  // run a LINQ select against the input

  // s is the current word

  return a.Select(s=>{

    // nibble digits from start of the

    // current word to build up the

    // current prefix length

    for(m=n=0;

      s[m]<58;

      n=n*10+s[m++]-48);

    // append the prefix from the

    // previous word to the current

    // word and capture values

    // for the next iteration

    return

      p=p.Substring(0,l=m>0?n:l)+

      s.Substring(m);

  });

}

Scala, 226 129 bytes

Still golfable, but at least it's here. Thanks to ASCII-only's work here (and for the Groovy answer).

(s:String)=>{var w=""

var c=0

"(\d*)(.+)".r.replaceAllIn(s,m=>{var a=m.group(1)

if(a>"")c=a.toInt

w=w.slice(0,c)+m.group(2)

w})}

Test zone in footer here at Try it online!