Technically, a spouse who moves to another country is still supposed to pay child support. Legally, though, it can be very hard to get him or her to pay. The reason is that many countries do not have standing laws and agreements saying they'll help if a spouse flees over the border to avoid payments.
There are officially 195 countries in the world. Just 15 of them have agreements with the United States regarding the enforcement of child support. There countries are Australia, the Czech Republic, Canada, El Salvador, Hungary, Finland, Ireland, Northern Ireland, the Netherlands, Poland, Norway, Portugal, The United Kingdom, the Slovak Republic and Switzerland.