The Federal apparatus (U.S. Dept. of Health and Human Services, Administration for Children and Families) runs an interstate information sharing network and sets guidelines for State and County Child Support Enforcement agencies and provides oversight and funding for those agencies.
Unless there's an existing jurisdictional decision for an order of child support the Fed's have no authority to act on a claim or author a decision for a child support order.
Collecting Child Support from a Parent in the United States - U.S. Embassy Bogot?, Colombia
However, some States do have specific reciprocal Child Support Agreements with foreign Nations and once an order for child support in one of these Countries is obtained the custodial parent can seek enforcement of that order at the State level without being required to meet the residency requirements of that State.
What this means is that once a judgement is passed in that State for an order of Child Support the obligor will be liable to meet their financial obligation in that foreign country and vice-versa.
Parent to Pay Lives in Another State or Outside of the United States
Massechusettes is one State for instance with several of these agreements. The difference is that an outside claim would be scheduled using that jurisdiction's formula and NOT the Massachusettes formula for determining payment. In other words a Child Support order from another Country for the equivalent of, say $40/ Month would be used rather than the State's formula which would probably result in an order of $400 - $1,000/ Month.
Reciprocal Agreements
Each State is different. The Fed's don't have an enforcement apparatus, just guidelines, information and State initiated collection referrals which are forwarded to the Internal Revenue Service.
So, in short a deadbeat Dad or Mom from Mass. could relocate to the D.R. with no fear of being forced to pay their court ordered child support. Likewise, a Dominican could move to Mass. and do the same since there's no reciprocity between the two jurisdictions. BUT...if they left any property behind or have any income source (savings, pension, social security, investments) back in the home jurisdiction it can and will be attached by the Fed's upon petition from the requesting State and if the obligor were to ever return to their home country (either) they would find themselves in prison until the arrearage, penalties and interest (usually in the hundreds of thousands) is repaid. Through Federal guidance, all 50 States have child support collection reciprocity and the Fed's can order the authorities in New Mexico (for ex.) to apprehend a delinquent obligor, all property and accounts for prosecution in Massechusettes (for ex.).