Having thought about this some more, I think I was a bit hasty to suggest the three tags should be synonyms. As @JanMurphy has pointed out, all three tags have a slightly different purpose.
My only issue is that it is often hard to say which tag should be used. A good example of an unclear case is Finding records of emigration/immigration from Germany to South Africa? In an ideal world, perhaps we would duplicate the tags and add immigration and emigration, but there are only room for 5 tags, and other tags about location and date are (in my opinion) more important.
So I would suggest keeping all three tags, but on questions where there is some ambiguity and use of any of the tags would be accurate, to default to immigration, because that tag is the most widely used and understood.