Sand dribbling out under baseboard. What could cause this?
Solved:

The answer was: airflow.  The sill plate does not sit firmly on the foundation, leaving a gap.  Differential air pressure in the basement and the room results in a small wind, which brings dust and (yes) bits of sand from the concrete or mortar.

A little bit of spray foam (using the gun, not the instant-clogging single use cans) solved the problem.