Apparently this was United States policy way back at the turn of the last century. Millions came from all over the world. But in those days the US was not crowded. In those days the world's population stood at around 1.5 billion - today it is in excess of 7billion! There are very few nations in the world that do not regard themselves as 'full up.'
As we know, China took the radical step of limiting one child to each parent couple. Huge problem! They all wanted boy babies. So now we have a China with far too many young men for the young women who live there.
Third World nations, in fact any nation which does not have a social-safety network of aged and disability pensions, will continue to have large families so that they have a 'family safety net.' That is, plenty of younger, fit, family members who will look after them in their old age.
The way I see it, the only real way to keep massive migration from flooding the 'Advanced economic countries which do provide social services, is to lift the poorer countries to the state their native dwellers feel happy and contented to stay where they are. When nations reach a comfortable parity in living conditions the urge to start a new life in a new land will likely drop by 99%