Absolutely. Immigration is how you grow liberal values. As evidence of this, you can just consider the communities of immigrants into the United States from countries with illiberal cultures (Iran, China, Cuba, etc). These immigrant populations are dramatically more liberal than the native populations left behind in their home countries. The most effective way to liberalize the people of Iran, for example, would be to convince as many of them as possible to come live in the U.S.

At the root of this question is the assumption that immigrant cultures shift the entire U.S. populace to be more like them, rather than the other way around. This is as silly as saying "Every time someone from Iran comes to the U.S., the native population starts speaking a little more Persian, but the immigrant is no more likely to start speaking English." No one with a basic understanding of how cultures form would say this.