For about the third time tonight I've clicked away from an enemy and my champion has moved forward.

That didn't happen to me, but earlier today, I distinctly recall ulting towards someone (as Ahri), and watching her ult away for no reason... I'm pretty sure that lost me the kill (though we still won the game, if I recall correctly)...

There are two main reasons I'm aware of that can cause this.

You could be suffering connection problems of some kind. You click near an enemy as you go in to cs or harass or whatever, then click away to back out. First click went through but the second got delayed (lag) or dropped entirely (packet loss).

The second would be input/display latency. Say you move your mouse to the right, click to run, then move your mouse to the left to be ready with your skill shot. If there is latency so the game reads your click a half second later you may have already moved your mouse to the left side of the screen and hence the game registers you clicking to run towards your enemies. Display latency would be the same but instead you see the mouse on the right when it's actually already moved to the left and just hasn't updated the screen yet.

I suspect you might be suffering the former. I was getting allot of packet loss from Riot earlier today after the patch hit. It seems to have cleared up now, thankfully.