Or do the leading edges appear to move (because water condenses on the leading edge, and evaporates on the trailing edges)?
If you look at fast motion films of clouds, it looks like this is what is going on.
The air that the cloud is in moves, and carries the cloud with it. The edges of the cloud may well be shifting at the same time, but the cloud as a whole will move with the air.
Some clouds move with the air they are in and some don’t.
Lenticular clouds like this appear to be stationary but the air is moving through them very quickly.
those are gorgeous! 
You can see how UFOs can be reported when these are around.
Perhaps both of these things happen at the same time, if the conditions are right. I once watched with fascination as this was going on:
One bright sunny day, I noticed that, if I looked towards a certain point in the sky, there were clouds to the right of that, but the sky was crystal clear to the left of that point. Yet there was a breeze, and the clouds (in that clouded portion of sky) were clearly moving from left to right (as seen from my point of view). Yet the boundary between the clear and clouded portions of sky seemed to NOT be moving. How could this be?
As I watched for a while, I became clear: At a specific small area in the sky, clouds were forming (out of “thin air”) and proceeding to move from there towards the right (from my point of view). That spot in the sky where clouds were forming was, however, stationary. So, towards the left, the air was warmer and cloud-free. But the air was moving from left to right, and cooling as it did, and at that specific stationary spot, the air was becoming cool enough to form clouds, which continued to move with the air towards the right.