Hungarians are distantly related to Finns linguistically, and also migrated from what is now Russia, but Hungarians look pretty much like their present-day Slavic or Germanic neighbors in Europe.
Like I said, I generally agree with your assessment and assertions, both for Hungarians and for Finns, though their languages are only very distantly related to each other, even if grouped together. And I had heard even that the relationship was not so much of vocabulary, as shows the relationship among Indo-European languages, but of syntactical and grammatical structure - word order in sentences, that kind of thing.
Maybe I could have worded it better: ‘of Siberian origin’, or ‘shared with autochthonous people of Siberia but not with their European neighbors’. I’m looking for the link now.
Abstract We have analysed data of three European populations speaking non-Indoeuropean languages: Hungarians, Lapps, and Finns. Principal coordinate analysis shows that Lapps are almost exactly intermediate between people located geographically near the Ural mountains and speaking Uralic languages, and central and northern Europeans. Hungarians and Finns are definitely closer to Europeans. An analysis of genetic admixture between Uralic and European ancestors shows that Lapps are slightly more than 50% European, Hungarians are 87% European, and Finns are 90% European. There is basic agreement between these conclusions and historical data on Hungary. Less is known about Finns and very little about Lapps.
It's from Cavalli-Sforza, who is someone I think that still has a good reputation, though this is from 1990, and maybe the 'less is known about Finns' has been updated since then. http://www.ncbi.nlm.nih.gov/pubmed/2221031?dopt=Abstract
I just searched 'Siberian ancestry of Finns' in wikipedia and that's the first thing that hit. Like I said also, you might be able to find what I'm referring to (or maybe mis-referring to) faster than I can.