A divorce judgment involving custody and support will be enforced across state lines. There is reciprocity between all 50 states. However, there is a process to make this happen. An attorney can aide you in registering your judgment in Missouri. An copy will need to be acquired. There are however requirements that must be met and you can discuss those with an attorney. You must have adequate contact with the state of Missouri.
Additionally, some very proactive clients ask if you can do this as a sort of housekeeping matter prior to any real issues arising.