that is an important point...

While we / they all speak Indo-European, that only proves the diffusion of the languages, but not really the movement of populations that is implied when you affirm that all Indo-Europeans share a genealogy if you go back far enough in time.

Some Indo-European migrations can and have been proven - particularly the Celtic and Germanic peoples whose migrations shaped the course of the Western Civilisation. But it just can't be argued universally. Genetic study of migrations has rarely been able to, as older scholars used to expect, match populations and their migrations with language diffusion, least of all for languages and language families spanning a large space. Mostly, because populations easily adopt a new language without having actually been displaced "genetically", just as did most of the Europeans conquered by Rome in adopting Latin which became Romance or all the migrants of quite the diverse origins to the USA who adopted American - it just doesn't follow that Walloons, Galicians, Sicilians and Moldovans all have their ancestors in Rome...

Pan-Aryan theories of the style and historical era of Nazism fail because of that. Of course, even outside Nazism, it still was the general opinion in that time that language and language diffusion followed racial / genetic lines and reflected actual population patterns of migration, substitution and displacement. But the genetic study of Finns and Hungarians has since disproved that they are not "genetically European", whatever that is supposed to mean, anyway.