Part of them, yes. Finland became a Grand Duchy, part of the Russian Empire after they beat the Swedes in the Finnish War 1808-09. During that time Russians moved in to trade. One of the most wealthy families in Finland nowadays is originally one of those who moved in during that time.
After the Soviet Union went to even more shit and eventually collapsed, a lot of them came in as immigrants, including from the Ingria area which was taken from Finland after the Winter War. I think there was a law passed that granted everyone from that area a citizenship even though they were born in Russian territory and spoke only Russian.
Nowadays they come in to work in construction or in berry farms and those kind of jobs where there's not enough native workforce to be found. Most are in for temporary stay but I'd imagine some find a way to stay in. Also the rich elite somehow finds a way to acquire Finnish passports, like the Timchenkos and Rotenbergs who bought the Helsinki Jokerit hockey team when the team moved to play in the KHL. I'd imagine there's many other businessmen who have done the same.