Question:
Do US states have any such system? Have they tried to discuss it? Texas, for example, has been sending migrants by bus to democratic-ruled cities. But is there a formal framework?
No.
Months before the American Revolution ended at Yorktown, Virginia 1781, the United States organized itself under the failed Articles of Confederation 1781-1789. This failed attempt at self government in the United States is rather similar to the modern E.U. in it's properties. It created a rather loose collection of independent states with a weak federal government unable to directly raise taxes and where important decisions require unanimous agreement. Primary federal revenue came from import levy's or unreliable funds donated by the States by unanimous agreement; otherwise the federal government had no income.
While this was an important stepping stone in the United States it was ultimately replaced by the United States Constitution 1789. Under the Constitution Article I, Section 8, Clause 4: The Federal Congress has full authority over immigration, and does not require consultation with individual states. The State's representatives speak for the states in Congress and reach a majority consensus rather than a unanimous consensus required by the Articles of Confederation or the current E.U.
And while few Americans who desire swift action on pressing needs would consider the U.S. Constitution an expedient or efficient governance. It has proven with effort to be workable, which is better historically from an American vantage than our experience under the articles of confederation.
Making this leap of organization in the United States was rather difficult. It required years of failed governance, federal financial defaults, and a rather clandestine campaign by the leading and most admired man in the country at the time, George Washington.
Europe which has millennium more history and ingrained nationalism than the United States faced has a more difficult job in making this adjustment to their collective organization. They might decide making such a leap is not in their individual state interests.