How's this for a late reply?
The answer would be 'both'. Some worked in Scotland temporarily to earn passage to the USA, others stayed, or couldn't earn enough to continue their trip. There's an interesting article about what happened to the Lithuanian names of those who remained in Scotland, specifically Glasgow in this article. Sometimes people also went back to Lithuania. Maybe they stayed months, or a few years...long enough to have a child or four. Did you look for Anna's birth on http://freebmd.org.uk/cgi/search.pl ?