This gets complicated as everything to do with DNA seems to be. Not that long ago scientists decided to see if men with the surname of Cohen [or its variants] really did have a common ancestor. It turns out, in many cases, that
they did. However, their exact same Y-DNA was also borne by males who had other surnames. Why this is can be explained by the fact that Jews did not have last names until fairly recently in the history of the world and had to purchase them or acquired one by some means or another. Therefore, one can conclude that many Jewish men knew they were cohanim, descended from the priestly caste, and took the name Cohen, accordingly, but other of their co-religionists had lost track of their ancestry or simply preferred another surname. Also, in more modern times, some Jews changed their last names to ones which were not so obviously Jewish.
Moreover, not all men with the surname of Cohen belong to the same DNA haplogroup and those, also, share Y-DNA with other Jewish males with different last names. If one man is of a different haplogroup than another, there is a good chance they are not related or that, even if they once had a common ancestor, the Y-DNA of their lines became altered at some point. There are several means by which this can happen, one of which is that a man who has no son adopts one, perhaps a child of his wife from a previous marriage or one of her relatives. If that man had a brother who had natural sons, there are now two collateral branches of the same family with the capability of propagating themselves into the future, but the scions of each having a different Y-DNA.
Statistically, however, it has been determined that the most likely haplogroup for a cohen who is really a descendant of Aaron the Priest is J1[with a refinement of J1c3, also called J-P58], this haplogroup definitely having its roots in the Middle East. Since Moses was the younger brother of Aaron and the Bible does not say they had different fathers, the following is his predicted 12 marker DNA, matching that of the modern J1 Cohens who were tested by Family Tree DNA:
/12/23/14/10/13/15/11/16/12/13/11/30/ with additional markers seeming to be:
"And Amram married his father’s sister Jochebed, and she bore him Aaron and Moses; and the length of Amram’s life was one hundred and thirty-seven years." Exodus 6:20