You are nicer than me - I was thinking of an invasion and military takeover. No vote needed.
If their government cannot provide for its people although one of the richest countries in the world and instead must support the efforts of their illegals to invade America to rob from our wealth - that appears to me to be a "soft invasion of our country".
And, since we cannot come up with a way to stop the flow - we just make them one of the states and instigate our laws to clean up the corruption there and bring about an effective state government able to provide for its people.
Problem solved. Few miffed egos, but - hey, we don't intend to be brutal to those people, we intend to give them the glories of America which apparently they desperately seek as the flow indicates.
And the wealthy corrupt ones might learn that you keep your corruption in your own borders.
(I know - too radical and a waste of time after I said - don't waste the time of serious problem solvers here).