Is a welfare state with open borders sustainable?

Is it a good idea to combine a welfare state with open borders? Milton Friedman the famous libertarian economist suggested that the US could benefit from an open border policy (he later modified this view). But he assumed that we would not hand out benefits to new immigrants at the rate we give them to existing citizens.

The left is a strong advocate of an open US border that allows any immigrant to cross into the US and become a citizen (and of course vote for them). They also happen to be an advocate of a strong safety net (code for welfare state).

So a welfare state (with public housing, aid for dependent children, state and federal disability insurance, old age pensions, unemployment insurance, school breakfast, lunch and dinner programs, public health insurance, public dental programs and public education) is one thing if the population is stable. But how expensive would it be if we were once the richest country in the world and then opened our borders to anyone that wanted to cross (legally or not)?