I used Europeans as an example but what about Polynesians who may have emigrated to America in the 1700s aboard commercial ships? What if they had children with native Americans or native Mexicans in the colonial period? Testing their descendants for the influence of external DNA would be most problematic.
I believe this is a valid study, but I also believe that some sort of precautions would need to have been taken to obtain genetically pure native DNA.
It seems to me that I have read of evidence from one of the Polynesian islands of some genetic material that seems to have come from South America...as if the Polynesians got there and brought some people back with them.