I come from a Finnish family, and since I was a little boy, I'd been told my ancestors came from Mongolia. I'd also heard that when they first began emigrating to the US, some (implied to be my relatives) were sent "home" to Mongolia. However, upon further research, this doesn't seem to be true, and none of us look particularly East Asian. Where did this idea come from?
I don't know if this comment will be removed, but the idea of Finns coming from Mongolia is a false idea dating to the 19th century, when scientist were really into the idea of different races and superiority of the anglo-saxon race. For example, a French scientist, who never visited Finland, wrote how Finns were like the Sami people, and how Finns looked like asians with our 'slanted eyes'. This is extremely weird for a Finn, because we look nothing like asian or the Sami people. Finland does have a small minority of Tatars in Eastern Finland, but it isn't really notable.
Shortly put, it was a misconception that was made up by people who didn't even visit Finland in the 19th century.
It also has to do with language. As you know, Finnish isn't related to most other European languages, so it presented a puzzle for linguists about it's origins. There was a thought it might be part of the east Asian language family.