I think there are two distinct approaches here. One, is to fully pursue a graduate degree outside of the US.
There are some graduate programs and fellowships available that will cover the cost or significantly reduce the cost of tuition, in countries outside of the US. These are great opportunities to get a global education and also expand the realm of career options. One example is free tuition for international students in Germany (via DAAD) or a Fulbright Scholarship, or Rotary Global Grant.
The other approach is to pursue a graduate degree at a US institution and participate in a study abroad program while pursuing the degree. Those options may be more limited, but as part of your consideration you should inquire about this when investigating graduate programs.
Employers value international experience gained via study abroad. It will certainly be an asset in most multinational, global corporations. It's important to learn how to unpack that experience in your resume and in your interviews. Good luck!