Did people in Siberia/Asia back then know that there was another continent across the land bridge? Some people must have made multiple trips right?
What evidence do we have if any?
Vajda (2008) proposes a shared common ancestor between the Na Dene languages of North America and the Yenesian languages of Siberia. While the theory has its critics, most notably Lyle Campbell ( his response ), many linguists find the evidence quite convincing. According to Vajda'a Dene-Yenesian hypothesis, the Yenesian languages arose as a result of a back migration by people already in North America. Thus, this is exactly the example you are looking for.