There are degrees of integration. People from Älvdalen, should they choose to, could move to Stockholm and change their dialect (one of the ways you know it's a dialect, is that they understand you much better than you understand them). It's been that way for a long time.

And from what I understand Älvdalsmål is, like all dialects, getting rounded at the corners and getting more understandable to other Swedes.

Even dialects that sound incomprehensible at first, if you're a native speaker you'll get used to it quickly. The difficulty of Älvdalska is superficial, it's actually very close to what you're used to, so you'll learn to understand them and they already know how to understand you.

Sami is completely different. It takes a long time to learn. Go back 150 years, and very few Sami would be able to move to the capital and pass as Norwegian or Swedish, their accent would give them away even if they did know the majority language. Go back another 50 years, and they may simply not have been allowed to even try to pass in many places (as I recall, the first Sami priest in Norway, Anders Porsanger, was rejected by his Trondheim congregation. He was simply too weird for them, even though he was highly educated and of course spoke excellent Norwegian).