Thanks to everyone for all the suggestions. Just to follow up:

There is a return on the third floor, at the top of the stairs.

Air is coming into the rooms quite strongly, so that isn't the problem.
I lit a candle and held it over one of the vents, and the flame got
brighter and roared. I held it near the return, and the flame leaned
noticeably toward the return, and its smoke got sucked right on.

Strange thing: while bringing the candle closer to a vent made the flame
brighter and more violent, bringing it closer to the return caused it to
get weaker and weaker. Holding it steady next to the grill, I saw it
barely stay alive, and finally die. Why would that be, I wonder. I would
think that fast moving air would be fast moving air, whether headed into
or out of a duct.