How to convince my PhD advisor to keep working with me if I am switching university?
It sounds like you should probably not expect your advisor to continue as your advisor. In choosing to leave the program (and, indeed, the country), you have relieved him of responsibility.

Advising students is difficult and demanding work. It is much more difficult, and less rewarding (for both the advisor and the student), when done over correspondence. Usually when advisors agree to do this, some combination of the following two factors are in play: (1) the student is near the end of their Ph.D. studies (and already has a thesis problem); (2) the advisor left the university (either temporarily or permanently) and the student didn't want to or couldn't follow them.

I don't know your exact situation, but I would recommend looking for a new thesis advisor in your South American university, while still trying to salvage a productive relationship with your current advisor. Even if he is no longer willing to assume formal responsibility for advising your thesis, he might still be willing to suggest open problems to you, offer advice, or be supportive in other ways.

One course of action that might turn out well for you would be to investigate opportunities that will be available to you at your new university, and schedule an appointment to discuss with your old advisor which of them he thinks it would be most valuable to pursue. 

Good luck to you in any case.