How does the "Special Orders" special event card work?
It's the dispatcher ability minus bringing people together.  For example, you can use it to move the containment specialist or (with a cure) the medic through cities to remove cubes.  We've done this to get double duty out of the medic late in the game when we were racing the board.

You are using your own move actions (and any constraints or benefits they bring) to move the other player.  (You spend your own cards for flights, for example.)  The other player's abilities are relevant if they are effects of moving, but not if they involve actions.

To address your cases:


Generalist moving the medic through a cured disease: cubes are removed; gets to move five times.
Medic moving the generalist through a cured disease: no effect (the medic didn't enter the city), four moves.
Troubleshooter moving ops expert by direct flight: Troubleshooter reveals the correct card rather than discarding it (his ability); but doesn't get to use an arbitrary card if at a research station (the ops expert's ability).
Someone moving the dispatcher, attempting to move a third player: no, you don't get access yourself to the other player's special abilities.  (But you could just move that third player instead with this action.)


See this thread on Boardgame Geek, particularly the comments from Tom Lehmann, the co-designer of On the Brink.  (Thanks to @tremby for pointing this out in a comment.)