There are distinctive Sumerian figurines from the Early Dynastic period. One of the early arguments for indigenous Sumerians was that these figurines were also in the Jemdet Nasr, Uruk, and Ubaid periods. Since the Sumerians according to Kramer came after 3250, I'm looking for evidence of these older figurines. I did a quick image search for Sumerian figurines from the Uruk and Ubaid period, and founding nothing like the Sumerian guys (shaved upper lip, hair in bun). Are there Sumerian figurines from before 3250?
This is from 1932, by the way. The Sumerian problem was later addressed by Joan Oates (Oates 1976).