When you share a child with someone who lives outside the United States, it can bring about some complications for custody arrangements. The situation will involve orders and laws from each country that could become confusing.
The good news is the American Bar Association explains the US courts and foreign courts typically will honor orders from each other. The tricky part is you will have more control if you file custody claims in the US court.
Filing for custody
If the other parent files a custody case in a foreign country, then you will have to deal with that court. It could mean traveling quite a bit to handle court needs. If you can file ahead of the other parent, then you can at least control which court the case plays out in.
Hague Convention
The Hauge Convention treaty helps ensure a parent cannot take a child to a foreign country and forbid the other parent from having access. Not every country is a part of the treaty, but it does cover most of Europe and allied nations in Africa, South America and the Middle East. It helps to stop parental kidnapping. You may need this if the other parent left the US in hopes of keeping you away from your child.
Details of your case
It is important that you establish custody as soon as possible for your children. You want to set the ground rules so that if something happens, you at least have a legal basis for resolving the situation. Sharing custody with a parent who lives in a different country may be difficult. But you do not have to endure problems and issues as long as you know what to expect and how the law works.