I'll comment more if I have time, but the headline points out what I think is the most notable point. This kind of Y-DNA diversity doesn't happen overnight. It takes generations. This points to a diversity of ancestry among discrete groups of "barbarians" in Europe that is greater than usually appreciated and a more cosmopolitan culture among these groups than is often appreciated.
Also, while the Hungarian example is usually considered an extreme counterexample of demic diffusion of language, with language shift secured despite minimal genetic introgression, there is lots of Uralic Y-DNA (see also Eurogenes) in this ancient DNA sample from Hungary.
Endre Neparáczki and his colleagues have just published a paper entitled: Y-chromosome haplogroups from Hun, Avar and conquering Hungarian nomadic period people of the Carpathian Basin. They analyzed the Y chromosome haplogroups of several individuals from 8 cemeteries in Hungary hosting Hungarian conquerors but also individuals from the previous Hun and Avar periods. The authors also tested some autosomal markers associated with eye, hair and skin color, and lactose tolerance. The authors obtained results for 46 men including three from the Hun period and 14 from the Avar period.