I imagine the lower relative temperature of the well shafts would cause cool fresh air to sink, and the heated (by humans and animals) stale air would rise up and escape via the mentioned shafts, much like a traditional passive heating system in an old house. As long as the area doesn't get too humid (unlikely in that area) then the latent heat of evaporation would generate more than enough temperature delta to setup quite strong air currents, more than enough to keep the air fresh in the caves