Considering this is a pre-historic event, you may not find much in the way of answers here.

Try /r/AskAnthropology/

Native Americans are mainly genetically related to Northern Eurasian population groupings that roamed the Northern Eurasian and North American (seemingly overlapping with megafauna distribution such as that of the Mammoth) where population groupings that migrated to North America had a higher ancient East Asian admixture. These Northern Eurasian population groupings migrated all over the place, to the Pontic Steppe, mixing with Caucasian hunter gatherer groups forming the Yamnaya and eventually into Europe. There seems to be another migration into Europe into Finland at some point.

But long story short, the current descendants of the 'Native Americans' who 'stayed behind' were people who now live in central and north eastern Asia. The only significant genetic contribution to these populations is probably a higher percentage of East Asian DNA.

If you want, you can look at pictures from native Siberians and compare them to Native Americans and Inuit populations, they do look quite similar not without a reason.