T. Even if you somehow managed to get past these roadblocks, dad can head to the courthouse the second you leave and get a court order for the child's return.
You can get a court order for the child's return but a US court order is toilet paper in other countries, not recognized and not enforceable, unless you have their courts check it out and agree to comply with it. That's why you need a really good international law attorney in international divorce cases.
With countries that have no extradition treaty to US, your goose is almost cooked at that point.
In my case, my STBX is a foreign national from an eastern European country that has no extradition treaties with US and isn't really on great terms with US. If STBX took my S4 there, and eventually got them to give him citizenship, you can be assured that country would not extradite their own citizens, a woman and a child, even if caught. So what if their father is a US citizen. You can go there, find them, alert local authorities and they will do nothing and I seriously doubt the US embassy can do anything to help either if there is no extradition treaty. Now with France or UK it might be a different story, if they extradite. I heard a woman kidnapped kids to France, they caught her and she lost all custody. But it really depends on the country.
China does not extradite, Philippines is the same way which is why it's a favorite place for criminals to run to. Check which countries signed the Hague convention.
Best to put your minor child on a non-fly list with the state department. The nutjob in my case is determined to leave and I have a feeling will find a way eventually. She was writing questions online, explaining how to run - and how to hide (e.g. change your name, change child's name, etc.)
