Some theories about the Lost Tribes speculate that they contributed to the Kurds ancestry. I really don't want to get into all the usual arguments on that.
What I first want to say here is it shouldn't surprise us there might be a correlation between descendants of the Medes and descendants of the Lost Tribes. II Kings 17:6 says many of the deported Northern Israelites were taken to the Cities of the Medes. Some suggest Mountains of Media was the original text, but I reject theories that say the text is corrupted, the basic outcome is the same though.
Jerome in his commentary of Hosea Chapter 1 said.
"The Ten Tribes of Israel inhabit to this day the cities and mountains of the Medes, as their fathers did a thousand years before."However I also want to comment on how the fact that some Kurds have the Y Chromosome that indicates descent from Aaron is used as evidence. Descendants of Aaron did NOT exist among the Exiled Israelites. We are told specifically in the account of the Kingdom's split in 1 Kings 11 that the Levites fled South when Jeroboam fell into Idolatry and then he created a non Levitical Priesthood.
The reason the modern Samaritans have an Aaronic Priesthood (or one claimed to be Aaronic anyway) is because as Jospehus records a brother of the High-Priest in Jerusalem in the time of Alexander The Great married the daughter of the governor of Samaria and then with Alexander's support built his own Temple there. Their local traditions later imagined a more distinct and ancient lineage for their Temple and it's Priesthood, but The Bible doesn't support that.
Two of the key Kurdish cities today are Nisibis and Irbil. Irbil was Arbela in the First Century. Both had large Jewish populations and were among the Mesopotamian cities where Jewish uprisings happened during the Kitos War in the reign of Trajan (Along with Edessa and Seleucia). That is likely when Kohen families came there.
Arbela was the capital of Adiabene. Adiabene in the Talmud is called Hadayb and is identified with Habor of II Kings 17:6 in Kddushin 17a. Where some of the Northern Kingdom exiles were transported.
Josephus records how Helena the wife of Adiabene's king Monobaz I converted to Judaism, around the same time so did her son Izates. A Jew named Ananias was their teacher in Judaism. Izates and Monobaz II his older bother both ruled the Kingdom. Izates died before 60 AD. Monobaz II assisted the Jewish rebellion against Rome in 66-70 AD, we know nothing about his fate or history after 70 AD, or indeed the Kingdom at all till the reign of Trajan. Helena and Izatez were buried in Jerusalem.
After both the 70 and 135 AD Diasporas I'm certain this was a place many fleeing Jews would have felt was a safe place to go. I follow up on this and speculate on Lost Tribes connection for Adiabene here.
Modern DNA studies have shown that modern Jews are close cousins to the Armenians and Christian Kurds. However Arabic Kurds are not so closely related, at least not as far as direct mater-lineal and pater-lineal lines of descent go.
Some researches though interpreting that DNA evidence see Kurdistan as the place of common origin and the Jews as the offshoot.
I believe the Ur that Abraham came from was Urkesh not the Sumerian Ur. Urkesh, Haran, and and the City of Nachor were all in what is today Kurdistan. Laban we are told had sons, so his daughters aren't his only descendants, Laban descended from a brother of Abraham. Nachor was the name of both the Grandfather and a Brother of Abraham. Another son of Abraham's brother Nachor was Chesed, I think possibly the ancestor of the Kasites who were also the Kasidim/Chaldeans.
Later these descendants of the siblings and cousins of Abraham mingled with the deported Northern Israelites, and Jews who came there following the Captivities of Judah. Also the Kingdom of Osroene who's capital was Edessa was founded by Nabateans, descendants of the firstborn son of Ishmael.