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If there are no orders regarding this in your custody order she can take the children where ever she chooses unless you get a court order preventing her from taking the child from the state.
In the end she may still be allowed to move if her family lives in another state and can give her and her child emotional or financial support that she needs. She could also be allowed to move if she can prove that she can better support the children by moving.
In order to stop her you will need to file a petition with the courts preventing her from taking the child out of state until a modification of the previous order can be done. At that time the judge will choose to either let her move or request that she remain in the home state.
Make sure you double check your custody order as most judges make a notation in the order prohibiting the custodial parent from moving without the consent of the father or the courts.
As far as child support you are required to pay it regardless of her leaving state. Child support and visitation are two separate issues and have nothing to do with each other.