I disagree with the above since that would render conversions certain provided a missionary station and time. If we went historical, I'd make the whole conversion thing apply only for pagans. I cannot think of an example where non-pagan populations were actually converted. Conversions in the sense that province switched from one religion to another all seem to have involved displacement of people.
Albania? Goa?
Not to mention the huge to and fro that happened between various Christian sects. Displacement of people played a part, but that wasn't the whole story. Consider for instance the conversion of England from a Catholic country to a Protestant one, in which the state played a huge role in forcing the new religion on the populace. Not many English Catholics actually emigrated to the continent.