I replaced the battery, and shortly after that, I noticed that the passenger window wasn't going down when opening the door. I did the sequence to reset the automatic windows, thinking that was the problem. The window rolls up and down just fine, but when you open the door, the window doesn't move down that inch or so. You hear the motor make a noise, but no movement. If you push on the window with your hand, the window will move down - Which leads me to believe this isn't an electronics problem, but some mechanical problem that just coincidently happened during the battery swap.

Any ideas on where to start?

 

Try cycling the top up and down a few times, followed by the window reset by pressing the window switch up and down. See if any change.

 

Some others seem to think it's the window regulator that's shot.. Fun.