Per the advice I got here, I decided to use three sanders - one for each grit - to sand the many small parts of my current project.
Per my own stubbornness, I ignored several suggestions for how to immobilize groups of pieces for sanding and went with my own idea; a four-sided "corral" made of thinner stock than the pieces to be sanded.
It's working pretty well so far, especially after I made a small tweak on the second run.