I'm confused about whether you want to live in one place in each Continent for a few months, or you are planning to travel around. If you are traveling around, then why just one country? Are you looking for a cheap place or the best place?
Just to throw things out there, if I were going to live in Latin America for a few months I'd choose Antigua, Guatemala (I've heard great things about Colombia for expats but haven't been). For Asia I would choose Hoi An, Vietnam (runners up Luang Prabang, Laos, and Chiang Mai, Thailand). Europe is harder as I haven't traveled there extensively since college, but Croatia is my favorite place to visit so I'll pick Dubrovnik.