I don't practice in GA but it sounds to me like it could be a case for the police to be involved. Many states if not all have criminal codes that include custodial interference. If you have a valid order, try calling local police and seeing if they will assist. Many deem these issues as civil issues and they don't want to get involved. If so, you have no choice but to file an enforcement or contempt action. You should consult local counsel.
If the children are in danger in any way, by all means involve the police. Otherwise, I disagree with that course of action. In enforcing your rights, you should always be mindful of the negative impact on the children of law enforcement involvement. It is scary. Refusal to return children will often result in strong action from the court, and will end up being a really bad idea for your ex.
As to what circumstances for an emergency pick up ... that's an awfully broad questions. Why not say what circumstances exist that you believe would make that necessary.