I've also noticed that auto-surveyed flag ships and automated mining ships wait to their next turn. For instance, you set them to automated or auto survey, come back and when your Turn button is green, they will still have turns left! They aren't using their turns effectively, it's a waste.
There are few different things going on. As nullspace mentioned ships on auto-pilot (usually) move at the end of the turn when you press end turn, but there are a number of situations where things don't work as they're supposed to.
This has been a problem in v1.2, v1.3 and v1.4. I haven't played DA yet so I don't know how it works there. What rev of the game are you playing?
When things work properly, the ship (or fleet) moves when you first set the auto-pilot destination but thereafter only moves on end turn. This should really make no difference since your auto-pilot movement occurs before any of the AI's ships move, so if your colony ship could beat an AI's colony ship to a planet if moved manually it should still beat the AI's ship if on auto-pilot.
Another thing that I've seen happen is that if the distance is large (I'm talking from one end of a gigantic galaxy to the other) the ship will take the auto-pilot destination but never move. If you move the ship manually until it gets closer to it's destination, then at some point auto-pilot will start working normally. I'm noticing this on v1.4 but I've never seen this on previous revs.
Another thing that can happen is that if you save the game and later reload it, you may find some ships have simply 'forgotten' their auto-pilot destination. This has been happening on all revs AFAIK.
The final problem is related but involves the auto-launch function to a rally point set for the colony that produces the ship. In this case again for no apparent reason, ships auto-launch but just sit there by the planet that produced them. If you go back and check you'll see there's no destination set for the ship even though the rally point is set for the planet. This also happens on all revs that I've played.
All of these problems seem to be related to the number of ships on auto-pilot and the size of the galaxy. Although all of these issues can occur randomly, they are far more likely the turn immediately following a load game. I've just gotten into the habit of moving every ship by hand on the turn following a load game.
I've heard anecdotal testimony that there is an auto-pilot timeout that will cut off the auto-pilot movement phase if it takes too long. This may cause the problem to occur more or less frequently based on the performance of your system. To me this is probably the biggest problem with the game. Functionality that isn’t reliable isn’t much better than no functionality at all. Auto-pilot and auto-launch to a rally point are great features that can significantly ease the micro-management pain in this game, but only if they work reliably first time, every time, otherwise they’re just useless baggage. I’ve been hopeful that this gets fixed in DA.