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Is caste based on birth? Is it inherited? Do any scriptures support birth based caste system?

Yes, caste is based on birth. There is scriptural support for it.

Scriptural support:

Manusmriti (10.5).- In all castes (varna) those (children) only which
are begotten in the direct order on wedded wives, equal (in caste and
married as) virgins, are to be considered as belonging to the same
caste (as their fathers)

Āpastamba (2.13.1).—‘Sons begotten by a man who approaches in the
proper season a woman of equal caste, who has not belonged to another
man, and who has been married legally, have a right to follow the
occupations of their castes.’

Viṣṇu (16.1).—‘On women equal in caste to their husbands, sons are
begotten who are equal in caste to their fathers.’

Yājñavalkya (1.90).—‘From women of the same caste as their husbands
are born sons of the same caste.’

Baudhāyana (1.17.2).—‘Sons of equal caste spring from women of equal
caste.’

One thing consider is that the actual genetic makeup of the 4 castes are different. Brahmanas have Nadis in their body situated in such a way as to receive the vision of the Lord. Kshatriyas are genetically strong and physically fit. Vaishyas have a mind and body conducive for conducting business. Shudras are naturally pessimistic, lazy, simple-minded, and materialistic. That's how their bodies were constructed.