Multiple reporting of derogatory information on the same debt is both proper and common.
OCs report derogs that occur under their account. If referred or sold to a debt collector, the deb collector(s) can additionally report, even though their actions were based on the same debt. If a judgment is later obtained, additional reporting of that fact can be added. The commonality of the same debt does not prevent the reporting of multiple derogs that occur along the road to obtaining its payment.
Owing child support collectible in one state, with subsequent shifting of the obligation by moving to a new state, similarly results in two parties having authority to report your obligation on the debt.
FICO scoring does not include Child Support tradelines, so there should be no impact on your FICO score at all.