For the moment consider only Vishnu Avatars.
Two Avatars, परशुराम & वामन were Brahmins,
One Avatar, राम was Kshatriya
One Avatar, कृष्ण was Yadav, (OBC)
Three Avatars कूर्म, मत्स्य & वाराह were animals,
One Avatar नरसिंह was a hybrid or humanoid? .
Now my submission is All वर्ण share common origin ie गोत्र.
गोत्र started with originating ऋषि , THIS would make all ऋषि पुत्र।
3 animals and 1 hybrid skew the वर्ण system, which वर्ण do we place them?
Coming to human Avatars.
At वामन 's time we probably had Brahmins who continued with their original task of study and teaching and others. (nowhere is Mahabali referred to as Kshatriya, which he is not, being of Brahmin ancestry). At this point only one + वर्ण exists.
By the time of परशुराम two वर्ण were established even though they shared common parentage - Brahmins dedicated in Shastras and Kshatriyas involved in Administration. This might be two वर्ण era.
By the time of Rama, Kshatriyas were rulers/administrators, Brahmins were scholar/teachers and then we had others. All sharing same गोत्र but doing different tasks. I would call it 2+ वर्ण system. Wonder which वर्ण does हनुमान fit in, he is a Monkey (वानर) with a prehensile tail, he is a unmatched warrior and जनेऊ धारी पण्डित as well?
By the coming of कृष्ण, वर्ण system is well developed. कृष्ण is a यादव (OBC), but there are marital relations between Yadavs and Kshatriyas, wtf?
Probably वर्ण wasn't the barrier it is made out to be.
Not an Avatar but still pertinent -
All 10 Sikh Gurus being Sodhi(s) or Bedi(s) trace their lineage to लव & कुश and they are खत्री s. This shows a large amount of social mobility where in descendants of राम are no longer Kshatriya.
Guess we can assert that Avatars formed themselves without considering caste, embodying themselves as to whosoever existed.
