I am thinking that until all the child support, and where people could live and how far children can be away from another parent is worked out, no one can go anywhere. I would speak to a lawyer PRONTO

If nothing is filed with the courts, either spouse can do anything with the kids. One spouse can move them. The other can show up without warning and take them back home again. There are no legal restrictions. Neither the mother or father have any more legal rights to the kids than the other.

Once divorce papers have been filed, there is usually a stay ordered that prevents either party from disposing of assets or moving the kids until custody has been settled.

I don't recommend that either parent take their children out of the other parent's life by moving away. If that happens, the parent remaining can take legal steps to have the children returned.
A good lawyer ensures that the divorce agreement specifies the children remain in a geographic area (which could be as small as a school district) unless the court allows a move.