If you're a separated or divorced parent who has concerns that your co-parent will take your children outside the country without your permission (and possibly not return them), there are steps that you can take to help prevent that.
The U.S. State Department has something called the Children's Passport Issuance Alert Program (CPIAP). If you enroll your child in that program, the State Department's Office of Children's Issues will notify you if your co-parent requests a passport application for your child, as long as your child is under 18 and qualifies for U.S. citizenship.
As long as you're the child's parent and/or legal guardian, you can enroll a child. Your attorney may be able to do it for you.
If your child has a U.S. passport when you and your spouse separate or divorce, you can still enroll your children in the program. This allows the State Department to notify you if a renewal application is submitted. However, it will not prevent your co-parent or a family member from taking your child outside the country. The State Department has no mechanism for doing that.
Further, CPIAP doesn't apply to foreign passports. Therefore, if your co-parent and/or your child has dual citizenship, your child may be able to obtain a passport from the other country.
You can seek to include a provision in your child custody agreement that prohibits your co-parent from taking or allowing your child to be taken abroad without your permission. Therefore, if he or she does so, that parent is violating a court order and can face serious penalties.
Source: U.S. Department of State - Bureau of Consular Affairs, "Child Abduction Frequently Asked Questions," accessed April 27, 2018