I’ve now seen this on two different cars of the same model & year and I couldn’t find a reference to this specific resolution on the Internet so I figured it would be worth posting.
The symptom is simple. The power sliding door on one side or the other will stop working. I’m guessing that this problem is shared between the Dodge Caravan and the Chrysler Town & Country since they share the same parts for this mechanism. Some people on the Internet say that the problem went away after they had the dealer flash the BCM (Body Control Module). Other people had to get an entirely new BCM before the problem disappeared. We never went down either of those paths since I had a simple quick, although temporary, fix. If you open up the fuse panel under the hood next to the battery:

Then remove, count to 5 and then replace the following fuse:

The doors should work again for a while.
With the first van that had this problem, we went for many months just pulling and replacing the fuse whenever a problem occurred, and it would usually only stop working every few weeks. A while later we ended up replacing the battery after the van wouldn’t start one morning. Since the door issue was only an occasional problem, I never really realized that it never happened after we got the new battery.
Fast forward a few years, and we ended up having to replace the van, and we opted for the same exact model & year. When we had the “new” used van for a couple of weeks, the left power sliding door stopped working one day. We were on a trip so I did the quick-fix fuse pull & replace, and the door started working again. Within a week, the van wouldn’t start one day. A jump start got us working again, but I took the van to Auto Zone to have the battery checked, and it failed the load test. I replaced the battery on this van, and we haven’t seen the doors stop working again!
My guess is that the Body Control Module (BCM) that gets so much attention has a failure mode that is supposed to disable the sliding door motor if it detects a short or a stuck electric motor, but that when the van’s battery starts getting marginal, it trips into this mode unnecessarily.
UPDATE:
Since this post still gets a bit of traffic, I wanted to add a bit of an update. After many more years of wear, we have had a few more challenges with these sliding doors. If you get to a point where most functions work, but one doesn’t then you’ve probably managed to actually break a wire in the harness. This can manifest as something like locking or unlocking no longer working, or open/close not working, while other things still work. If the break is still not quite complete, this can be an intermittent problem for some time.
I tried finding and splicing the broken wire one time, and it only lasted a couple of weeks since there is a fair amount of strain as the doors open and close. I ended up just replacing the entire harness and I was pleasantly surprised how easy it was. First time (passenger side, more use) took me a bit over an hour. When the driver’s side failed I managed to replace it in about 45 minutes. You can get the parts from Amazon (links for 2004-2007 model): Passenger Side [747-311], Driver’s Side [747-310].