I imagine it’d depend on the circumstances of the disappearance. Assuming there’s no evidence if foul play (b&e, blood, property damage, ransom note, etc.) we can rule out a criminal investigation.
If someone is on the cruise alone and simply decides not to return to the ship after a stop (in port) – I’m not sure anyone would notice right away and the ship would simply sail along.
Now if someone reported someone missing (spouse, family member, traveling companion) – a search would probably be conducted onboard, local authorities (closest, most recent port) would be alerted (who would start whatever their standard missing persons investigation is).
If it’s suspected the person went overboard at sea, the coast guard would be called in to conduct a search.
If it’s determined that the person is neither onboard or fell into the ocean, I suspect the ship would continue on it’s cruise (after some delay) and the investigation would be left to local authorities. The missing person’s traveling companions (if there are any) would likely stay behind.