After Batu Khan's forces destroyed Rus and Kipchaks/Cumans, around 200k Cumans escaped to Hungarian steppes. Mongols asked them to the Hungarian King pretending Cumans 'belong to Mongols, since Mongolian army invaded and destroyed Cumania. Not to be given back to Mongols reportedly Cumans mass converted even to Hungarian catholic Christianity.
Hungarian king decided not to return them. It was year 1241, Hungarians were next to be destroyed by Mongol armies.

During the Russian campaign, the Mongols drove some 200,000 Cumans, a nomadic steppe people who had opposed them, west of the Carpathian Mountains. There, the Cumans appealed to King Béla IV of Hungary for protection, in return for which they offered to convert to Western Christianity. A mass conversion would enhance Hungary’s prestige with the pope. Moreover, the Cumans pledged 40,000 warriors, experienced in the Mongols’ mobile steppe warfare, to Hungary’s defense. Béla gladly accepted the offer, but many of his nobles distrusted the Cumans. His decision gave the Mongols an official excuse to make Hungary their next object for conquest.

After holding a council of war in Przemysl in December 1240, Batu sent an ultimatum to King Béla IV. Word has come to me, he wrote, that you have taken the Cumans, our servants, under your protection. Cease harboring them, or you will make of me an enemy because of them. They, who have no houses and dwell in tents, will find it easy to escape. But you who dwell in houses within towns–how can you escape me? Rejecting the ultimatum, Béla sent heralds throughout Hungary carrying a bloody sword, the traditional symbol for a national emergency, to rally the nobles and vassals to the kingdom’s defense. http://www.historynet.com/mongol-inv...f-liegnitz.htm
Hungarians look mostly like their neighbors and if you see some Turkic effect it's them.