For everyone except the top 0.01% of earners, “income inequality” is actually skill inequality.  To see why, consider two thought experiments.

1. Suppose the US labor market starts in equilibrium with 10m skilled jobs done by 10m skilled workers, and 30m unskilled jobs done by 30m unskilled workers. Suddenly, we figure out how to train 10m unskilled to become skilled. We now have “too many” skilled, and “too few” unskilled, for the available jobs.  From Econ 101, what must happen to skilled and unskilled salaries?  Obviously, they get closer to each other.  Income inequality goes down.
2. Suppose the US is in equilibrium with 30m unskilled jobs done by 30m unskilled workers. Then we bring in 10m unskilled immigrants. The new arrivals can perform only unskilled work, but they demand work of various skill levels (doctors, lawyers etc). Thus, even if the new arrivals’ overall impact on the labor market is exactly neutral, they will create a shortage of skilled workers, and a surfeit of unskilled.  Thus the price of unskilled work must fall, and for skilled work must rise.  Income inequality goes up, unless you can do #1 above — train lots of unskilled to become skilled.

This argues that immigration policy and education policy are inextricably linked.  Unless you are able to train unskilled workers, you must not admit more unskilled workers, or you will end up with rising inequality and political instability.