Animal species will expand into suitable habitat nearby, certainly. Whether they cease to exist in their original range is usually a question of the habitat there becoming unsuitable for some reason or another - maybe through climate change, or increased predation, or because that species has changed the original habitat itself - which is what you seem to be talking about. That is usually a question of overgrazing or similar, and typically will be a cyclical thing: population boom leads to overgrazing, which leads to migration and/or population crash due to starvation, which then allows the food source to recover and rinse and repeat.

I am struggling to think of a particular animal species which has permanently changed the habitat of an area to the point where they couldn't survive there - other than human. There are plenty of plants and microorganisms though. That is the whole basis of ecological succession.