Firebird doesn't use as many Russian folk tunes as Petrushka but here are two that I know of.

The 'Round Dance of the Princesses' is based on the song 'The Vain Suitor' for which I have both the dots and the lyrics in English. Should I post them and what about copyright issues?

The finale of the Fire-Bird is based on the song 'The Larch Tree' I have the dots for this and some but not all of the lyrics (in English - don't have original Russian, I'm afraid).

Hopefully someone more knowledgeable about this will post but, if not, perhaps this could be a start for you?