But as you say, the rapid transformation of the societies of the groups that migrated at the time is very hard to comprehend, but also the changes of those where they migrated, in cases where it was more of a fusion than an invasion / displacement.
I think it is a very interesting point to offer to those in Europe minimizing the current migration waves (often for noble reasons of reducing racism and countering far right ideologies). It's no use denying that welcoming tens of millions of people from other cultures over a few decades has and will have very deep social and political effects.
Well no one really knows because most of this comes from the writing ruling classes. If we're talking Middle Ages and the Modern Age, France offers an interesting example : absolute monarchy by divine right came very late (17th cent) and was almost never fully accepted. The system relied greatly on religion (which kept one's standing within a social class), weak origin stories (the Franks came and conquered the Gallo-Romans/ my ancestor fought in the crusades, which is why you're a serf), and a very strong system of local rights and privileges (I'm a legitimate ruler because I keep your right to hold poultry markets on Sundays, which the next village doesn't have).
The divine dimension you refer to was a constant effort in France and England (where the king was not a religious ruler) well described in "The Royal Touch" by Marc Bloch.
I know other examples of relative ethnic stability: most of Northern Europe broadly has the same ethnic geographic boundaries as 1500 years ago (excepting the slavic / proto slavic things), the Basque, the Sardinians, Bretons and Bearnese, Tirolians, Kabyles, Laz "Greeks" (vanishing since WWII).
By some measures also Egypt (coastal cities excepted) and Iran.