"How Hard Should I Fight This? And, What is the Hague Convention?"
You should fight HARD. Under the Hague Convention on the Civil Aspects of Child Abduction, signatory countries may seek to have a child returned to his or her country of habitual residence. This is very complicated, and you will need knowledgeable family-law counsel to guide you through how the treaty operates in each country (who speaks French or English, which are the official language of Hague treaties, plus the native language). This is a time-consuming and expensive process.
It is important to note the Hague Convention only applies to signatory countries. Many countries in the Middle East, South America, and Africa have not signed on. Also, children are defined under Hague to be those of 16 years of age or younger. There are selected legal remedies that may apply in non-signatory states.
If international abduction is a concern in your case, the first step you should take is to obtain family-law counsel to prevent the child from leaving the United States. There are several tools available to lawyers and litigants to prevent this. These include a court order to surrender the particular parent’s and child’s passport to the Court or attorney.
At Ciyou & Dixon, P.C., we have worked with such delicate situations in the past and have retrieved children from other U.S. states and foreign countries. Where possible, we attempt to work on planning restrictions to maintain the child in the United States to avoid this situation.