There are some countries that won't recognize the DO for full-time practice. Meaning that you couldn't necessarily move there and practice as a physician in their system.
But for medical missions, there are no restrictions. DO's participate in Doctors without Borders etc. without any restrictions. This is because when you do a medical mission, you aren't trying to obtain a license to practice medicine in the country you are volunteering in, but rather offering volunteer services on a temporary basis.
In short, the only way a DO can limit you internationally, is if you want to emigrate to one of the countries that doesn't recognize the degree, and get licensed to practice medicine there full-time.
It should also be pointed out that there are plenty of countries that have the same restrictions for US MD's as well. It's a political thing, rather than a "training" issue.