We have had a few over the years that lived in various European countries while being based in the US...we currently have a guy living in Greece, and another one in Dubai.
We have had guys also living in Thailand, Philippines, etc.
Needless to say that this becomes difficult for bidding, families, and even your pocketbook since the company will NOT buy a ticket for you from anywhere but the 48 US states and Alaska...even Hawaii is not included in the Gateway travel plan.
There are plenty of opportunities to JS on our own aircraft from FRA, HHN, RMS, etc from Europe and from HKG, PVG, and ICN in Asia and make your way to the US.
Couple of the guys I have talked to end up giving up significant number of days off just so they can live in Europe and Asia. They even end up working back-to-back schedules from one month to the next.