The evolution of the name Ashkenaz is distinct. It had nothing to do with DNA. It was a name for someplace up north, and when Jews stumbled into Central Europe they wrote home and tried to explain where they were.
All Jews came from West Asia, which includes Turkey, Syria, Mesopotamia, Eretz Yisrael, etc. It isn’t like there was a pure strand of DNA in one particular place. People moved around, frequently. Just note the various place the אבות lived in during a mere four generations. What the DNA has shown, that is significant and was very disappointing to secular Jews (who don’t like being Jewish), is that Ashkenazim are not descended from diaspora converts but are from the same “stock” as all the other Jews (so much for various theories that we are really Turks from Central Asia, or some other such ancestry untainted by Jewish blood).