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People did move back and forth across the bering strait well into historic times. It's not what this article is about though, which tries to look at shared morphology across geographic partitions to tease out the likely origins and timings involved. In this case they (very tentatively) identified coastal and inland origins that match up with other numbers and do not match up with beringian standstill hypotheses.

It's also worth emphasizing that Indigenous nations (first nations not being a sufficiently general term) don't have consistent views on this matter and don't usually identify specific dates or timelines.

You can identify specific positions advocated by indigenous individuals. For example, there are indigenous people who argue indigenous heritage in the Americas predates anatomically modern humans leaving Africa. You can also find indigenous people who agree with academically-accepted ideas about ethnogenesis. You can even find people who agree with both of these ideas simultaneously, similar to how you can find Christians who agree with consensus theories on human evolution and also identify the garden of Eden in the middle east somewhere. Rather than speaking about "first nations" as some sort of homogeneous mass, it's better to identify specific positions and talk about those instead.