In my US state, a parent who is owed child support will have a court order to that effect, and the ultimate remedy is a petition for contempt of court, which can result in the offender being jailed.
In between, the party who is ordered to pay can have his driver's license taken, his paycheque garnished, and can be ordered to prove he is actively looking for work if he pleads unemployment. A judge can stipulate that such a person show evidence of any number of applications the judge wants, and cannorder frequent updates on the job hunt.
The person can be fined too, and can be ordered to pay interest (currently 9%) on back payments ordered. Fines can be taken out of a paycheque. Someone who pleads self employment and next to no income can be ordered to show business accounts/ tax records. Again, the threat of jail time for contempt of court or a separate prosecution for perjury hovers over the offender. If a non paying parent is a member of a licensed profession - lawyer, etc, their license to practice can be jeopardised by such misconduct.
A judge can also place a lien (subject to due process) on real and personal property of someone who owes child support - house, cars, land, etc. A judge can also freeze a bank account and can also divert any tax refunds from the offender.
Prosecutions are handled by the state's attorney/ district attorney (at county level). Leaving the state with the intent to duck court ordered child support payments will result in pursuit by the FBI and federal charges. States will track down offenders using all resources available to them, including PIs.
Losing the driver's license or the threat of losing it often works wonders. Driving without a license can have major legal consequences.