What compelled them to stay in such a harsh enviroment as Alaska/Yukon/Nunavut/Northwest Territories? Why did they stop moving after they cross the Bering Strait?
This is a fairly broad question but I will attempt to tackle it in separate parts:
(Edit: I should have mentioned that I'm answering this with an Eastern Canadian Arctic perspective with the Thule in mind)
Territory roughly to the south of the 60th parallel (where the provinces begin) was already occupied by First Nation Peoples (Inuit are not considered First Nations having crossed the Bering Strait approximately 1000 years ago). Moving south means encroaching on Dene/Cree/Innu/etc territory. Still to this day, Inuit and Dene are negotiating over traditional hunting grounds.
Moving south does not equal a better life for Inuit. Inuit hunt (depending on the group) polar bear, beluga, seal, walrus, etc. These species are rarer or non-existent the further south you go. Considering these animals are central to Inuit way of life, why would they move away from it?
Padlermiut, inland Inuit, often frequented near the treeline into Manitoba territory and interacted with the Dene First Nations. Their language (mainly based in Arviat Nunavut, not far from Churchill Manitoba) is very different from other Inuit language dialects in Nunavut. They share loan words with the Dene. Some may have gone south.
It could be argued that Inuit have been moving south for a while now (forcibly at first with residential schools then voluntarily). Young Inuit are moving to cities like Ottawa, Winnipeg and Edmonton for work or school. Some move back but many stay in southern Canada.
I would recommend watching a great documentary aired on CBC entitled the Inuit Odyssee which delves into some interesting theories of why the Thule (Inuit) migrated from Asia all the way to Greenland. Here's a brief description of that documentary http://www.cbc.ca/natureofthings/m/features/history-of-the-thule-migration
(Edit, added this point) 6. In the 1300s the Earth warmed up and Inuit of the eastern Canadian Arctic and Greenland moved north. This would indicate that they were following animal migration rather than search for warmer climates.