Is it usually the case that once you graduate from an MBA program abroad you get some sort of visa to look for work in that country? Are there countries (with good programs) that make it especially hard for you to stay after you've graduated? For example, countries like the UK (Cambridge Judge and others), which have made it harder for foreigners now to move (or stay) there. Is this something to take into consideration?

Thanks.

Yes. Visas are not the only factor,  since language skills are also a massive barrier. It's worth understanding what the barriers are. International mobility is actually very high for people who come to study a top MBA in the US because the language barrier is low. 

The UK has actually moved towards making it easier in recent years to transition from a study visa to a work visa. 

English speakers tend to have a couple of major choices. For long-term visas, the US can be very difficult. Canada, Australia, and New Zealand tend to have more liberal visa policies but perhaps different levels of career opportunities between then.