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Unless the Court Order is modified to say otherwise, it is the responsibility of the custodial parent (or guardian) to make sure the child gets to her visitation.
Although the child's wishes are one factor in determining the "best interests" of the child, the Courts do not give a 14 year old child absolute power to decide what is best for her. She might not want to go, but if the court has ordered it, she has to. If she doesn't, the custodial party can be held in contempt of court for not making her go.
That makes the issue more of a discretionary one. There is no cut and dried answer, but generally, you will not be in contempt for not making her go. You should file a Petition to Modify the order at some point though.
I can't give you a 100% answer, but generally, it is up to the parents to agree on something... you do not HAVE to put her on a plane.