the thing we often forget about the huns, and this is so important, is that they were really just an aristocratic elite. if they hadn't evaporated, they could have maybe been understood using renfrew's model. but, they left very scattered traces, mostly in german mythology. "hungarian" as we understand it as a kind of corruption of history; these people were never the historical huns.

when the huns attacked the empire, they did so using a very broad coalition of the willing that was composed of an assortment of german and iranian tribes, most notably the goths and the burgundians, and the alans. the huns themselves, famous for their archery, were not the foot soldiers. they just provided air power. it was white people that did the pillaging and slaughtering...

the mongols were something else, and separated by several centuries.  they famously upped and turned around for reasons that have never been clear to history, leaving an ad-hoc system of tribute that took eastern europe centuries to emancipate themselves from.

regardless, you can't describe these people as indigenous to eastern europe - for they are clearly indigenous to the eastern steppe, which is where they first appear to history as the barbarians that the chinese built a great wall to keep out, and were also eventually subsumed by.