Disclaimer: I know absolute nothing about this topic and this is from my recollection of reading wikipedia and soft-science articles
As I understand it, it should have taken a lot of time for full migration from North to South America; and while a thousand years seems more than enough for travel, apparently migration takes a long, long time - and the odds are that a fossil comes from a migrated population, virtually impossible to be from an odd outlier that traveled ahead (for whatever reason)