It doesn't. This does nothing to get people to pay their child support, and the people who can't afford to pay it aren't in any position to flee the country. This law isn't about getting them to pay child support, it's about stripping rights from the kind of people who can't or don't.

They're trying to require people to prove citizenship in order to vote. If this policy happens and that requirement becomes the law of the land, Boom, poor people poll tax. If you only think of each thing that's happening as an island that can't change anything else, you're going to miss the forest for the trees.

36 states require identification to vote. A passport is ID.

Gonna stop everyone right here and mention that ID requirements were not the norm until recently. It is, and always was, possible to prove citizenship without issuance of an ID (let alone a photo ID). Studies have shown that it does very little to reduce voter fraud and only disenfranchises vulnerable populations.

Secondly, elections are the purview of the states. Any requirements that give feds control of the voting process are unconstitutional. A federal passport is US property issued at the discretion of the government and can be revoked at any time (as shown in this article), and is therefore unconstitutional.