It's very confusing the way a language can be imposed on a preexisting population which once spoke a completely different tongue. Some studies show that the genetic makeup of the British people has not changed much in several thousand years, although first Celtic, then Germanic, was imposed as the prevailing language by Continental invaders. Then there are other studies showing that the Anglo-Saxons committed ethnic cleansing on the preexisting population.
BTW, the Mongol penetration of much of Scandinavia was pretty limited ~ they collected taxes and such (expensive furs and amber). At the same time there are all those folks around who have Type B blood eh!
What I think happened to the folks who invaded and conquered what we call Hungary (or Estonia) is they simply got genetically swamped. When they invaded they were able to outnumber the natives ~ which is why they were then able to maintain a long term presence. Numerous other Mongol (and other Eastern) groups came after them. Still, they improved local economic conditions so well they attracted Slavs and Germans to the area. In the end populations that started out speaking any number of Indo-European languages became Hungarian and Estonian speakers.
Give you an idea how this happens, I was looking up the genealogy of a French relative and found I could easily run it ALL THE WAY BACK to the 800s to a couple of Magyars named Álmos and Arpad.
Worth noting that EVERY royal house of Europe for the last 500 years has these two guys as their ancestors ~ and not a one of them look all that Asian!
One that intrigued me was the Hungarian claim that people of their cultural identity/ethnicity were the first permanent settlers of the Carpathian Basin.
Check that, the "Carpathian Basin" ~ although the Carpathian Mountains actually had Sa'ami language speakers right up to the Middle Ages. So, whatever is found in the mountains the Hungarians are right up to date in admitting that if it looks Sa'ami, it's Sa'ami and not Hungarian.
You have to be careful in reading the material about Sa'ami at any point in time ~ they were until just a few years ago thought to be part of a common genetic and linguistic pool of Finno-Ughric peoples who divided from their Finno-Altaic relatives in Asia as recently as 3,000 years ago. DNA tests demonstrated the Finno-Ughric have no relationship at all to the Sa'ami within the last 10,000 years (except maybe the Skolt with whom they share some ancestry among the natives in Latvia).
Many of the quick references in Wiki still reflect the Pre DNA test thinking. If you find yourself reading a reference to Finno-Ughric people in the Carpathian mountains, they are really talking about the Sa'ami, not the Finns.
Now, for a relevant point. Carpathia (the area in the vicinity of the mountains of the same name) covers parts of several countries. In the relatively recent past the Czar of Russia obtained control of most of the area and brought his own military forces and management team to the area. I think that is about 1806 or so ~ the Swedes had just lost a war to the Russians, and had to sign over Finland to the Czar among other places. Up to that time this was pretty much under the influence if not control of the former Swedish Empire.
I didn't know these Sa'ami were there, but I've been watching closely for the departure of large numbers of Sa'ami from Finland at about that time in history. Lo and behold some let, but not all by far. On the other hand, Sa'ami language pockets are not reported in the Carpathian mountains after the Russian takeover. Many Sa'ami from UNKNOWN PLACES show up on the American frontier by 1810/12. I'd presumed they all left from Finland and Norway to escape a major famine, but apparently a bunch left from Carpathia ~ speaking a dialect of Skolt Sa'ami (spoken by highlands people in the Eastern Sapmai who herd reindeer in Finland and Russia, but mined gold, etc. in the Carpathians). That particular language was the one I reported as having just expired (above).
Fellow I worked with had a fascinating genealogy. His 5 times Great Grandfather had been among the Cossacks brought from Russia by the Czar to take over the Carpathians!
His Grandchildren all ended up in America digging coal in Pennsylvania.