I think the best bet for a minecart for junk, would be to set it like a stockpile. Have it accept, for instance, microcline, within a certain distance, like 10 tiles away. Set a microcline stockpile beside the cart. Then it will generate a job of "load" where it grabs 10 microcline, or less if there's none nearby, and then the rider pushes or otherwise powers the cart to the offload, where it sits in the unloading dock's inventory like the way any finished good sits in a workshop. Once there, a dwarf unloads it to a nearby stockpile, or not, and if the drop off is full then it won't generate a new "load" job back at the source.
For that matter, every drop could have its own requirements. Like, have one drop that takes lignite, another that takes limonite, and a third that takes marble. It will try and stock each of these to 10, and when one drops below 10 it generates a load job back at the source, which causes a cart to be loaded and deliver the goods to each drop point in turn.
This would more or less replace stockpiles for certain things, but they should probably be treated like trade depots combined with stockpiles. A 3x3 drop point could hold 8 units, allowing it to hold 8 things and have 1 tiles left over for loading and unloading.
I just realized that this thread is about moving people, so it's not entirely accurate, but, moving people would be more difficult as they're all trying to pathfind and they'll be at different distances from the cart itself...