To help prevent international parental kidnapping, since July 2001 federal law has required the consent of both parents for a child under 14 to obtain a passport. There are exceptions, such as proof of the other parent's death, a divorce or family law court order granting the applying parent sole custody, or a court order specifically permitting the applying parent to travel overseas with the child.
Note that this applies to all foreign travel, including visits, not just permanent relocations. How well enforced it is at our borders with Canada and Mexico, however, will likely vary.
A well-written parenting plan will address the issue of international travel and passports. Typically, a Colorado parenting plan grants to each parent the right to visit abroad with the children during that parent's parenting time, as long as they provide the other parent with an itinerary, and plenty of notice (at least a month, so that parent has the opportunity to challenge in court the wisdom of a spring break jaunt to the latest war zone as unsafe).
If the parenting plan does not address foreign travel, a parent who wants to visit overseas must act well in advance, contacting the other parent for consent. If that parent refuses, then the remedy is to request permission to travel overseas from the Colorado divorce or family law court which has jurisdiction over your case.
This is especially important to military personnel who have primary residential responsibility over their children. Unless the original decree of dissolution allows that parent to travel overseas with the children, upon notification of an overseas PCS he/she should contact the other parent as soon as possible for permission. If that parent refuses, the custodial parent must request an order from the Colorado divorce or family law court to apply for passports and move overseas with the children.
U.S. Department of State. Web site with information on bringing children abroad, and the forms to obtain passports for children.