Also, your notion that certain people are inherently tied to certain places is only true if you accept arbitrary cutoffs for arrival time. The Lakota migrated from the Great lakes to the western planes in the 1700s, but it's hard to imagine them without their horses (which came from the old world) and bison (which they never would have hunted in ~Michigan). More time has passed since the arrival of the Spanish in Tenochtitlan, than between the arrival of the Aztecs (who probably migrated there from ~Arizona) and the Spanish. And of course, All the Native Americans descended from tribes that crossed the Bering straight from Russia, and every human's ancestors lived exclusively in Africa for a very long time. Anglo Saxons, Gaels, Norse, etc were constantly on the move within Europe before any of them crossed the Atlantic. Israelites displaced Canaanites before being displaced by Romans, then Arabs, then "coming back" after WW2. At any point in time, in any location on earth, the ethnic groups considered "native" almost certainly descended from people who were not considered native within at most a couple hundred years. Many white people can trace over 2 hundred years of ancestry in the new world. To say that's not enough to qualify as "native" is to have an incoherent conception of nativeness.
English doesn't help. I realize I myself used the term "Native American" while trying to argue that the term is out of date, but our language just was never equipped to describe these kinds of long term, highly chaotic processes.