Voluntary Human Extinction Movement

[The Architect Of Fate]

I mean that species can move closer to North and South poles. Only equatorial regions will probably become uninhabitable.

Nope, eventually, these effects will harm species even in the farthest regions of the earth. This also assumes these species can make it there. 

But people would move there only if those place will become MORE habitable.

Not necessarily. They may move there during a more habitable period only to then suffer the later drawbacks, by the way, permafrost affects things well outside its immediate region so even not living in the permafrost regions will be affected.

In case of Russia cold is the primary cause of low habitability

Now it is and in the future,  other factors will be the cause.

It was very positive for Sahara.

Under the climate conditions of the time by the method of warming that caused it. It can't be used as a comparison to now....