I remember reading about a tombstone from Ancient Rome, near Hadrians wall. It's a memorial to a woman named Regina, sat up by her late husband Barates. On the tombstone it said that Barates was from Palmyra. Now comes the question on what on earth Barates was doing in Britain thousands of miles away from Palmyra? This raises another interesting question. And my question is, would these people have looked odd in Northern England at the time? Would people have pointed at them and said that's the funny looking coouple from Palmyra? Or was this normal? This obviously speaks to the great mobility within the Empire, but do we know anything about how normal this was, this mix of cultures? Would it be normal for people living in Northen Britain to encounter people all the way from the opposite end of the empire, and the other way around?
The vehicle for moving people from one end of the Empire to another was the Roman army and its mercenaries, suppliers, and the like. Merchants and traders would also move goods and people along. You would expect merchants in major trade centers (like London) and soldiers and the like near military posts. While Hadrian's Wall might seem remote, it was also a defended border, and the soldiers (or civilians supporting the military) might have come from any part of the Empire, and some would have married and/or stayed locally when they retired.
So: you would expect multiculturalism in big cities or army posts. Not so much in remote locations.