Origins of Sumerians
The archaeological culture that is unambiguously related to Sumerians is known as Uruk period (4000-3000bc). Uruk period comes after the Ubaid culture (6000-4000bc)
Unfortunately we don't have ancient DNA from modern Iraq. But we have data from neighbouring regions. Both from West of Iraq and East.
The change from Neolithic to Chalcolithic are quite dramatic if compared to distant sources like CHG(63%) and Levant_N(20%). But it is quite unrealistic that there was two distinct migrations from Caucasus and Levant to South Zagros. More likely there was just one migration from a place that was between Caucasus and Levant.
Indeed if one use Hajji Firuz Chl (5000bc) from North West of Iran as a source of change then he gets quite good models in G25.
So we can assume that there was a mass migration from North West Iran or probably from North Iraq also to South Zagros sometimes after the Neolithic and before the 4000bc.
I must add that Iran_Chl do not show any evidence of migration from places like Turkmenistan, India or Siberia that has been proposed as possible homelands of Sumerians.
This migration from North is visible in Levant also. Chalcolithic era samples from Israel are close to preceding Neolithic period, but they also show an unambiguous migration from North Mesopotamia.
Distance: 2.8615% / 0.02861488
Those genetic results are not at all surprising. Because they fit well to the known facts that North Mesopotamian cultures like Halaf and Hassuna where quite influential.
Summing up all this. In the current stage of our knowledge the most likely origins of Sumerians is the Northern Iraq regions. And maybe even more Northern but not as North to reach lake Van. This is also in accordance with some linguistic data about the relation of Hurrian and Sumerian.