When a military dwarf (on duty or off duty, doesn't matter much) decides to perform a Pickup Equipment job, she'll drop the old clothing right where she's standing. This old clothing is still marked as belonging to her, so nobody else will touch it. If there is space in her personal quarters (either a cabinet, or open floor space), and if she is not otherwise busy, and is not a total slob, and if the stars are aligned properly, then she might move the used clothing into her personal quarters. If that happens, it'll probably stay there forever.
On the other hand, if she doesn't move the clothing into her personal quarters within a few weeks, the ownership will decay. The used clothing will become unclaimed fortress property. Then, someone else might claim it and wear it, or it might be moved to a finished goods stockpile, or you could mark it for dumping.
While the ownership tag is still there, there's nothing you can do to the clothing -- you can't dump it, can't assign it to another dwarf, can't move it to a different stockpile, can't stick it in a bin, can't sell it at the depot. There are things you can do to the environment around it, though. You can draw a stockpile underneath it, and turn on the Refuse category in the stockpile. This will cause the clothing to wear out quickly, eventually vanishing entirely. Or you could introduce water flow, which should push the clothing in the direction of the flow. Or you could introduce magma, which will typically destroy the clothing in short order.