Yes but as a hospital employee you would also know there is great value in maintaining inter-agency relationships. The EMS types have this down pat, and Police and Sherriff generally play nicely. If everyone starts whipping out their dicks to see who’s bigger, nothing gets fixed and the shit that does ends up fucked up. Customs will rarely interact with a hospital this way, but they probably come in and ask that these patients not be released without being processed by customs AMA or otherwise. Local PD will probably happily help keep tabs on any potential runners and has the authority to detain them if needed.
Oh, I can totally see informing some agency before discharging patients, but requiring the hospital to have a lockdown ward or something is a bit much.