Bob, you are right ... saying atmospheric pressure has
no effect is too sweeping a statement. In fact, it does have an effect in that as the atmospheric pressure outside changes (the onset of a storm for instance), the pressure inside the cave has to be brought into equilibrium and that will, of course, create a draught but only until balance is restored.
However, the draft will (usually) be very slow (because the atmospheric pressure change is slow) in this circumstance alone and it would probably only be detectable in a narrow part of the system where the airflow was severly constricted.
It would be better to say that atmospheric pressure can have an effect (albeit nominal unless the external pressure changes suddenly and to an extreme).
It is also worth noting that a "chimney" effect (well, a venturi effect) can also create a draft (I know this was mentioned earlier in the thread) but, again, you would have to have a significant wind at one entrance without the same wind at the other (thus, the venturi is created).
.... I am off to find some aspirin now .....

Ian