There are people who imagine earlier humans were tied to the land and didn't move around much. At the same time a few thousand Neanderthals could pretty much cover the entire landmass of Europe without much effort. They were pretty dangerous themselves and could kill, munch and eat their way across about any space they cared to.
Genes get passed in transit ~ that's the way it is. To think it didn't happen in the good old days is simply not realistic.
It took the Mandan more than a millennium to move from the Virginia coastline to the eastern Ohio River Valley to their final home in the Upper Missouri River valley because they were tied to the land through agriculture.
It took their more aggressive Sioux cousins less than half that time to cover basically the same ground and move even further west because they were not.
To this day, even some professionals are baffled by how the Sioux and Mandan came to possess certain iron tools before the white man arrived, even when it is clear that tribal cousins in the eastern Ohio River Valley and Michigan knew how to work with iron.
Human history does not take place in a vacuum which only begins to change when another culture comes in contact with them. I'm sure the same is true with the Neanderthal. That's why I'm skeptical as to whether DNA science is either as complete or omnipotent as its proponents claim.