As you can tell from my flair this is definitely not the right part of the world for me! But this is actually a medieval history question, at least, in the spirit of the “global Middle Ages”, which medieval historians like to talk about these days.

It’s really an archaeology question, since there aren’t really any written records for the Inuit in this period, so r/AskAnthropology might be more helpful. But I can try to explain.

The modern Inuit and related groups in Alaska, Canada, and Greenland did not arrive there until about 1000 years ago. They weren’t pushed into that environment by peoples further south, they already lived in that environment, and their technological and cultural adaptations to the Arctic actually helped them displace the earlier cultures that already lived there.

In archaeological terms the people who lived in the Arctic before the Inuit are “palaeo-Eskimo” or “paleo-Inuit” or “pre-Inuit” and the palaeo-Inuit culture that the Inuit displaced is known as the “Dorset culture”. The Inuit who moved east from Alaska about 1000 years ago are the “Thule culture”. The Thule culture might have straddled the Bering strait so they may be related to the Chukchi and Kamchatkan peoples of modern Russia (but maybe not, and their language don’t seem to be related - but again AskAnthropology and r/AskLinguistics might be better suited for that).

Of course, the groups that initially peopled the Americas may have also come across the Bering Strait - but that was thousands, maybe even tens of thousands of years earlier. The Thule migration is completely separate.

In Inuit tradition, the Dorset culture is known as “Tunit”, who were incredibly strong, maybe even giants. They hunted caribou and walrus and built homes and other buildings out of stone. Presumably, the Dorset people spoke Na-Dene languages, the same groups that now live a bit further south in Alaska, the Yukon, and the Northwest Territories. The Dorset culture was not as well-suited to the Arctic as the Thule culture. They don't seem to have had kayaks or dogsleds (and maybe not even dogs at all). They had harpoons and spears, but Thule harpoons were more advanced and better for hunting walruses and whales. It’s possible that the Dorset culture disappeared because they simply left when the technologically advanced Thule people arrived, but it’s also possible that they were displaced violently - Inuit tradition says the Tunit were peaceful and not used to warfare.

The Dorset culture was mostly or totally gone by the time of the “Little Ice Age” in the 14th century, when the Inuit also had to move south as their usual environment was too cold for both people and animals. That’s probably when they came into contact with the other Na-Dene peoples, and a couple of hundred years after that they also came into contact with European explorers and settlers. The Inuit may have already met Europeans in the 10th/11th centuries when Vikings briefly arrived in modern Labrador and Newfoundland, but it’s not totally certain if the Thule culture had reached that far west yet or if the Vikings encountered the Dorset culture (or other peoples entirely).

So in fact it’s the other way around, they moved there relatively recently, and pushed out other groups who weren’t as successful at living there as the Inuit were.

Source:

Robert McGhee, Ancient People of the Arctic (University of British Columbia Press, 1996)