I think it's still very much an immigrant church.
Today, since by far, most of the immigrants coming into the country are Mexicans, there are going to be many immigrants, however, in 1930's, it was totally an immigrant church, and in more than the sense that you are thinking.
The Catholic Church was always a church of pilgrims, people not of this world, hence always immigrants. After, Vatican II, they were pulled down to become of the World. There are hardly any Catholics left today that LIVE the faith, they are all of the World, no different in the way they live than anyone else.
If you want an example, just look around you and see if you can find somewhere a married for life couple with a large family. Families of 10 children were common among Catholics, however, today you are lucky if they are married in the Church, or even have children. In Austria, a Catholic country, less than 5 % of Catholics go to mass.
P.S.- among the USA traditionalist Catholics who attend the Latin Mass, and specially among the SSPX people, you do find those large families, even among the young, 30 and under parents. although I'm told that is not the case in Europe. Someone from there can comment on that.