There was definitely that. As one wave of immigrants became assimilated to whatever extent, another group was already on the waY in. In a way, this is part of the theme of the United States. Although there is grumbling and jostling to make room and for everyone to adjust, much of our strength has come from influxes of new ethnic groups with their own cultural strengths and weakness, new foods and devices. Even our brand of English is so eclectic. You can take an average newspaper page and circle words swiped from a variety of languages.
There's always friction as new groups arrive in numbers, but it almost always works out.. just in time for the next batch to start showing up.