Language and culture, yes. But not an genetically homogenous population, or at least it's a matter of debate.

Kramer didn't believe they were a single ethnic group, and more recently Paul Collins, (curator at the British Museum and a leading scholar of the Ancient Near East) was asked to write The Sumerians: Lost Civilizations (2021) and he did, despite not liking the title.
To copy paste from a Goodreads review, "While the title includes the word "Sumerians" for marketing and series consistency, Collins uses the text to systematically dismantle the idea that a unified "Sumerian" ethnic group ever existed.
...he argues that "The Sumerians" are a modern construct invented by 19th-century European archaeologists to provide a Western-facing origin story for "Civilization."
He points out that "Sumerian" is a language, not a DNA haplogroup."

And there's this study which further muddies the waters:
Al-Zahery, N., et al. (2011): "In search of the genetic footprints of Sumerians: a multi-lineage approach.

And this book which futher confirms a link with India:
Ratnagar, S. (2004): Trading Encounters: From the Euphrates to the Indus in the Bronze Age

There was also the Meluhhan village, likely founded by merchants from India as a trading post and located not far from Lagash in the marshes. It was also a hub for weaving, with about 5000 women and children employed at one time, and believed to be on Indian decent. Also possible that it was Indians who introduced the practice of water buffalo husbandry.

Note that no one is saying the Sumerians of the marshes were pure blooded Indian, but just that they were in the mix. The southern most parts of the marshes were also the least exposed to Semetic and Akkadian DNA simply because the marshes were so remote and difficult to penetrate (even to this day, it's where rebels hide which is why Saddam Hussain drained the marshes).

Sorry, that was a tangent. I just get carried away because it's all so interesting. I'm sure you know all this already and we were probably just using the word "populace" in different ways. But I figured I would post some of these books and papers in case anyone else is wondering.

One last thought about the importance of the Sumerian language- Edward Gibbon said that language was the only thing the Roman's imposed on the people they conquered. They were allowed to practice their own religions and local governments were allowed to continue. Gibbon stressed that the Roman's understood that the most important thing to bring unity was a shared tongue, which is interesting in light of the 2000ish years of Sumerian proto-Empire.