I believe you'd have an excellent shot at it. You could try to stop her from going and request custody based on the false information she provided. If not, you'll probably have to wait until something actually happens in order to seek contempt and modification, after you have it enforced. Your ex will likely find herself in trouble with law enforcement authorities, in addition to the family court. You should go to a lawyer sooner rather than later.
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Once she leaves the country with the children and keeps them there, it can become tricky. Most countries have treaties with the U.S. that will enforce existing U.S. custody orders. However, there are some that will not.
I would tell her no and make her take you to court to get the passports if you're concerned.
New State law states that in order for an individual to relocate themselves and/or the children into a location that is detrimentally hard on the other parent must file with the Court within 60 days prior to the proposed move. The Court will then hear the reasons for the move (work, income, what happens with the children, their friends, etc.) and determine whether the relocation is proper. If the Court finds the relocation is not for the best of the children, they can prevent the children from moving with mom. Since you share legal custody you have an equal right in their upbringing, and she cannot just move out of the country to avoid you. Doing so violated child kidnapping laws, and depending on the country (if they have an agreement with the US) can enforce the custody order and force her back. I would file an emergency petition (state when she is to leave) with the Court asking to block her relocation or prevent her from taking the children (as long as it really is not for the best of the children to move). I would seek an attorney, since time is of the essence and you want to do what is best for your children. Best of luck!
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