Numerous states and dynasties have fallen as result of local climate change, sometimes human driven. But civilizations tend to last much better.

The Sumerian culture as a living, everyday entity connected to a political system that identified itself as Sumerian came to an end, but the larger Mesopotamian civilization which Sumerians inaugurated survived and can be said to survive even today in a way. Civilizations are much more durable than states.

It's the "less developed" - to use a somewhat aged term - non-urban cultures living a more precarious existence that tend to fall victim to local and global climate change.