Alright, checked the bug list and found some that resembled both of these, but were not exactly the same.
I started a desert fortress in a new world and eagerly awaited the first caravan so that it would bring some fresh supplies. When it finally did arrive, I had my dwarves carry my trading goods to the depot I had constructed. The first problem arose when I ordered my broker to the depot to trade. Instead of going to the depot, he instead carried goods to the depot (with all hauling turned off), sat in on his usual meeting, and finally stood around no-jobbing while I tried in vain to get him to go trade by repeatedly turning the 'broker requested' status on and off before setting it so any dwarf could trade, which also did not work. Finally, worried that his inactivity would doom my fortress to death by dehydration, I dismantled the depot out of sheer necessity. Seeing that the wagons were still around, I rebuilt the depot to see if I could get trading to work and make sure I had gotten everything off of them that they were trading. After building a new depot a few tiles away, the wagons moved to the new depot, where the same bug persisted and my broker (or any other dwarves) still would not trade. Finally after turning the 'broker requested' setting off and on several more times the broker got off of his lazy ass and headed to the depot, where the trade screen commenced (it was empty, though, because I had already stolen everything).
Edit: Okay, an addendum to the second one. Apparently rebuilding the Depot will cause the wagons to leave for some reason.
This is where bug #2 starts: a few seconds after finishing trading, the caravan started to pack up to leave. During this time, though, I decided to tear down the depot and build it back in the original spot, so I scheduled it for demolition. A dwarf arrived more quickly than expected and started deconstructing the depot right before the second wagon had concluded it's 'packing up' phase, so instead of leaving, both wagons are now 'stuck' just outside of my fortress. Ugh.