It seems like they got displaced by the Akkadians.
The Akkadians (the Assyrians during the phase when the capital was in Akkad or Agade) moved in, or grew in numbers, and adopted the cuneiform writing as well as the epics and folklore of the Sumerians. The Sumerian language is agglutinative, and a language isolate, being unrelated to any other known language. Besides adapting cuneiform to Akkadian and other languages, the Akkadians continued to learn and use Sumerian, implying ethnic fusion.
The Sumerians referred to themselves as "the black-headed people" (apparently in reference to their hair color) and believed themselves to have arrived by sea as colonists, at a very early date. They believed that all cities were founded by the gods, and neither the names of their cities (including, I see here, Agade) nor the major rivers were Sumerian names, or Semitic, but pre-existing, apparently the few reoorded fragments of whatever earlier language or languages (and hence peoples or cultures) predominated in the region prior to the arrival of the Sumerians.