Highly skilled professionals in some fields (engineers, scientists, software developers, very senior managers etc) can often find a job with only English. These people are more likely to send their children to a foreign-language school, even if they have to pay for it, as they might not plan on staying forever.
Not knowing/learning the language also works for some jobs like cleaning or agriculture, if so many workers are migrants that everyone speaks Romanian or English etc. Also postgraduate students.
Plenty of people from the north who move south when they retire (e.g. Netherlands to Spain, Germany to Greece) move before learning the language.
How quickly people learn the local language depends. Slowly in Denmark or the Netherlands where English is almost universal and work is often done in English, quickly in France or Italy where the opposite applies. Slowly if you're 65 and surrounded by other Dutch pensioners.
From what I know of France, a child from the other side of France can also be excluded, but this is from French emigrants teasing each other. I haven't heard of similar problems in Denmark.
Growing up in London I didn't even notice the European immigrants in my class and could hardly exclude the 40% of the class who were descendents of South Asian immigrants, but this varies significantly around the country.
I expect compared to the USA there's much more reluctance to move when a child is 14-18 years old, unless they can be certain of a place in the French school in Copenhagen etc. But this is primarily because of the difference in exams and language.