Oh certainly, talking about genetics opens up a whole new can of worms. After all, each individual has their own genetic makeup, which could come from anywhere. In only a couple generations those genetic traits become part of their surrounding region. And although this happens in each region, it doesn't happen uniformly.

As for Hungarians, what I like to point out, are the roughly 1000 years preceding the last 1000 years of living in the Carpathian basin. During that time they were living a nomadic culture, along with Turkic peoples, in the plains north of the Caspian and Black seas. So while they may be originally Finno-Ugric, they are certainly Turkic as well. One thing doesn't exclude the other.

As for the Finnish language, I've had a couple of experiences where I heard a song (in some place far away from Hungary) that sounded like I should understand it. Upon closer listening, I still couldn't manage to grasp one singe word, but the intonation of the whole just had such a familiar ring to it. Of course I wrote it down, and eventually got the song myself... in fact, I'm listening to it right now, and still I keep having the same feeling. The group is called Loituma, maybe you know them.