People don't just come to a new country and just abandon their past. They bring their values, ideals, beliefs, problems and solutions. As the number of foreign people coming, cosmopolitanism increase.
Then you quickly reach a point where you can eat a raclette anywhere, soccer is the most famous sports, Hollywood is the main entity broadcasting entertainment. There is no point in traveling, because every place more or less lost their spirit.
And I am not talking about the fact that, if countries completely drop their requirements based on nationality, they will keep their "usefulness" requirements (who is able to take a job we are looking for)
What's going to happen? Well educated people are going to move easily. Further depriving countries of the very ones they need to develop their countries.
It will be a one way street. I don't imagine hundreds of thousands of Americans moving to eastern Africa anytime soon.