The Sumerians didn't come from elsewhere, nor did they go anywhere. They were the native inhabitants of lower Mesopotamia, and that's where their descendants remained long after Sumerian died out as a spoken language. I wrote more about this in Mesopotamians and Sumerians. Pay particular attention to the bit about the debunked "Proto-Euphratean" language substrate, as that was the primary piece of evidence for the theory that the Sumerians came from elsewhere.

You may want to read my response to What are the Assyrians and the Babylonians connection to the Akkadians? as well.

For general works on the Sumerians, see the Sumerians section of the r/askhistorians reading list.