As with almost everything in life, the point is in the balance. As you pointed our with a superb example, some immigration is beneficial for everybody, while mass immigration represent a problem.

I would just like to emphasize that the level of problem that mass immigration causes depends on the cultural difference between the immigrants and the natives. For example, if Irish people would massively emigrate to United States at some point, that wouln't cause as serious problem as if Nigerian people would massively emigrate to Poland, or vice versa.