I don't think it's a majority. But let's say there are 10 major categories from which converts come. I see a bell curve where protestants are over represented in western lands.
Your comment about Latin America is accurate. That would fall under my category of former missionary activity. I have first hand experience with rural South America and the work there. Many small towns and villages are predominantly Catholic. But these are largely isolated from the majesterium of the Church at large. Many members find appeal in the symbology and mystical aspects of Catholicism in such conditions. Greek and Lutheran Orthodoxy does not have a large hold in many of these areas.
What you say about the Islamic states is accurate. But the fact extends to the U.S. and western lands. The ability to argue Christianity vs. Islam is not really possible. Converts in western lands from ethnic communities and minorities have great impetus for a common social circle where people who are generally left out can fit in and rub shoulders on par in the theocratic arrangement. So, an immigrant farmer might be a social equal with people whom he wouldn't be outside the organization.