Are there any instances of Lord Vishnu or any of his avatars speaking in favour of either conduct-based or birth-based varna?
Lord Vishnu in the form of Krishna says in the Bhagavad-Gita:
मां हि पार्थ ब्यापाश्रित्य येऽपि स्युः पापयोनयः।स्त्रियो वैश्यास्तथा शूद्रासतेऽपि यान्ति परां गतिम्॥
English Translation: For finding refuge in Me, O Arjuna, even those who maybe of sinful birth, women, Vaishyas and similarly Shudras, even they reach the Supreme goal. (Bhagavad-Gita 9.32).
I think the very word paapyoni makes it clear that He accepts the birth-based varna system. Also, He did not protest at all when Arjuna spoke ill words about the Varnasankaras.
Yes, according to the Vishnu Smriti:
Viṣṇu (16.1).—‘On women equal in caste to their husbands, sons are begotten who are equal in caste to their fathers.’