This phrase is said as part of the script of a film where a CIA agent who is romantically involved with this woman Carol is traveling with her and apparently going on some sort of official mission where she can involuntarily help him with that mission.
So, that's what's his colleagues guess that he is doing: "He must be running Carol".
What synonym verb would you use instead of "run" in this case?
What's exactly the idea expressed here? Does it mean that the guy is taking advantage of Carol in some way?