It's both melee and magic, but it seems to build by type (for example, rdm+thf made them immune to magic (enspell) and melee after 15 or so mobs, but ranged worked after that just fine). It might be something weird like majority damage per kill (pup/whm and nin, they became immune to melee faster than ranged, which led to dead puppet >.>). Who knows.
But it does decay pretty fast, I farmed them for a week at one point, 2-3 eggs a day (after that damage tended to be 0), and the next day it was again tolerable. Not totally reset (enspell did 13-14 from start instead of 17 like it should've), but manageable.
As for their tp moves, those are just plain weird. At times they'd do Frigid Shuffle (the third ws, Yawn > Beak Lunge > Shuffle) from first mob on when damage was still fine, at other times they never did it, even when I was reduced to dotting them to death with bio and poison.