Is a child support order constitutional and lawful if the custody case was settled with a legally binding verbal contract and that contract was breached? For example:

I make the offer to allow full custody to my ex, as long as she cannot leave the state with the child without my express permission and she accepts in the presence of a judge on record.

If she leaves the state with our child, against my consent, is any child support order that she files for, in another state, valid? Will the breach of contract make her liable to court action against her? And if so, will a new custody proceeding be held and the child support order invalidated?