In many cases it is left to the discretion of the original agency involved, i.e. if military police were the first to determine a soldier is missing, they might search for him (possibly with CID) assuming he is AWOL or call in another agency if foul play or what are commonly agreed to be FBI matters are involved (i.e. specific counterterrorism, organized crime, etc.). As @Bluefreedom stated, the FBI’s broad reach and resources are better equipped to locate a missing person. Army CID, for example, does not have the size, presence or resources to go hunting for missing grunts.
As for the CIA, @Introverted_Leo—the Agency does have an Inspector General and a police force with limited domestic law enforcement capability, complete with employees classified as 1811 (Criminal Investigator). Bringing the FBI in to hunt for a missing spook might be problematic because of security clearance issues, so I think it would depend on the missing officer’s work specialty as to whether or not it would be handled in-house. My initial response would be to say that the FBI would be brought in and provided with a CIA liason, and given secret information on a strictly need-to-know basis.