Do you have a custody order or parenting agreement? What does it say about travel? How old is the child and what country is the child going to? Does the child have a passport?
You should go to the U.S. Department of State website and look at the section on foreign travel to see what that country requires for a minor child to enter the country. Different countries have different laws. Canada, for example, requires a court order or a letter from the non-traveling parent for a child age 15 or younger to cross the border by car. If the child is 16 or if the child is traveling by plane, then a passport is required.
For a child under 16 to obtain a passport, both parents have to give permission. You can place a block on your child’s ability to get a passport by filing a request with the U.S. Department of State through the Children’s Passport Issuance Alert Program (CPIAP). As the name implies, if you file a request with CPIAP, then if the Department of State receives a passport application for your child, they will follow up with you to confirm that you did actually give your permission.
If there is no court order in place regarding international travel, you may also file an action asking the court to prohibit the travel or require the traveling parent to deposit a bond with the court to insure the return of the child, if you have evidence that the traveling parent does not intend to come back to the U.S.