I tried to reply but Mudcat promptly went offline for two days before I could post it.

There is a discussion of tune families in Nettl's "The Study of Ethnomusicology" which you can read on Google. He covers a lot of other people besides Bayard. The point I hadn't thought of is that all these different classifications use different principles - Bartok's was nothing like Bayard/Bronson. In particular, linguistic aspects (metre, rhyme scheme) mattered much more to Bartok and those he influenced than they did to Anglophones. The Bartok-influenced study I've looked at most is Vinko Zganec on Croatian songs; nothing like Bayard. But the textual way of doing it goes back in the Anglophone world to the classification of hymn and psalm tunes adopted early in the 19th century; in secular music it was used on a massive scale in the French "Le Clé du Caveau" (you can read that on Google, at least the first edition).

Kodaly is a bit more like Bayard, since his big project was to unify families across the (perhaps rather dodgy construct of) historically Hungarian and related peoples, over many centuries. Janos Sipos is quite a bit saner although he covers an even wider range of cultures, right across Turkic Central Asia (and has some to-me-incomprehensible remarks about Native North American music which attempt to pull that in somehow). Sipos's work is mostly available free.

Nettl is pretty pessimistic about the prospects for unifying these different family classifications across national boundaries.