The EUROCONTROL example you give, allows controllers from all member states to work in Maastricht Upper Area Control (Netherlands) and provide ATC over Belgium, (part of) Germany, Luxembourg and the Netherlands. Many of the controllers there are not from any of these four countries so they are foreign air traffic controllers.
Also EUROCONTROL is only providing Upper Area Control above the four countries mentioned above. All member states have their own Air Traffic Control providers, some have more than one.
Within EU countries (note that not all EUROCONTROL member states are EU member states) we have freedom of labour movement. So for EU citizens there should be no official barriers to work as a foreign controller in another EU member state. Apart from the intra-EU rules, some ATC organisation within the EU are open to applications from outside the EU. For example the Dutch Air Traffic Control Organisation LVNL is open to foreign controllers. Applicants are not limited to citizens of EU countries.
ATC at airports is becoming a commercial business. The companies that you mentioned provide civil ATC at various airports around the world. Another example is ATC at London Gatwick Airport: the contract was awarded to a subsidiary of the German ATC organisation DFS.
Outside the EU I know that there are many foreign controllers work in the United Arab Emirates. They are looking for new applications at this moment as well.