We moved to a different State. How do we go about transferring jurisdiction?
Generally, if there is no pending litigation, you would file a motion in the court in the new State whenever you needed to. (There would be no need to file anything until you have an issue to bring before the court in the new State.)
When there is pending litigation (a case in progress), the original court will usually retain jurisdiction until the matter is settled. At that point jurisdiction may be transferred to the new State.