It sounds like those early Hungarians were sort of like the Cossacks, who may or may not have come from around Kazakhstan. When we visited Ukraine (back before all the present warfare,) we were told that many, many Cossack tribes were from that area, and were possibly descended from the Mongol hordes, allowing for several generations of interbreeding with locals. Of course, "Cossack" later came to be the term for various czars' professional thugs, who "kept the peace." The original Cossacks were all-male tribes with strict orders of horsemanship practice and deportment. They were celibate; those wishing to marry and have children were required to leave the tribe. (Or at least that's how it was described to us.) They said that the name "Cossack" was derived from "Kazakh" (Sacha Baron Cohen wasn't around to confirm or deny!) They said further that there were some who thought the name came as a corruption of "cassock," since the standard garb was a cassock-like robe. But since that was standard garb for most Russians at that time, it seems highly unlikely.
P. S.: Papa Elf is very proud of his "Praha" hat.