This has happened and is happening in Europe, too.
Many people are coming in, some of them don't integrate and cause problems, the center says it's not a problem and the left says let's have more of them.
More people are coming in, problems are getting worse (both real and imaginary), people are getting upset, the right realizes they can use that and they build their whole agenda or that and win the elections.
The number of countries this has happened in increases, so non-right parties need to rethink their strategy if they want to stop losing.
Europe is able to change political course much more gradually: the EU is 27 countries, and the EU Parliament is elected with proportional voting systems which leads to coalitions and compromise.
A 10% increase in 'right' votes means roughly 10% more influence for the 'right' opinions.
In the USA, a tiny increase in 'right' votes means 100% more influence.