Full Faith and Credit shall be given in each State to the public Acts, Records, and judicial Proceedings of every other State. And the Congress may by general Laws prescribe the Manner in which such Acts, Records and Proceedings shall be proved, and the Effect thereof.
Let’s say Joe and Maurice get married in Vermont. When they move across the river to New Hampshire, their marriage is still legal, even if their marriage wasn’t legal in New Hampshire.
Then they move to Nevada and live there awhile and they adopt a couple of children. Those children are their legal responsibility in all states. Maybe they wouldn’t be able to adopt legally in another state. That doesn’t matter. Joe and Maurice are still legally the parents of those children.
Then they move to Louisiana and unfortunately, their relationship deteriorates. They get divorced. The divorce decree, including property division and parental responsibilities, are legal and enforceable in all the states, even if other states have other laws.
The idea behind this is that if you enter into a contract that is legal in the state in which you entered into it, it is enforceable in all the states.
Ultimately, the responsibility for how such inter-state differences are enforced are the responsibility of Congress.