Quick answer
Interstate support rules can allow cooperation between agencies, but the authority to change an existing order needs separate review.
Do now
Gather every order, the issuing court, payment records, and a history of where the parents and child lived. Tell the agency about other cases. Ask about registration or interstate enforcement rather than opening an unrelated new order without disclosing the old one.
Watch for
Support jurisdiction and custody jurisdiction are different. Moving does not automatically transfer either case. Avoid inconsistent orders by identifying the existing case at the outset.
Official sources
NY Courts — Starting a Case in Family Court, DC Courts — Parentage and Child Support Branch, DC Code § 16-4602.01 — Initial child-custody jurisdiction.