I wonder if lifting the plywood is what is actually drawing in the ash. Assuming your smoke box is essentially a six-sided cube, the plywood lid makes up about 17% of the total surface area. Lifting it will create a momentary vaccuum in the smoke chamber - the mometary vaccuum is equalized by pulling air in from anywhere it can - the fire box is an open passageway where air will momenarily rush through (carrying ash along with it) to equalize the vaccuum created by lifting the lid.
I'd simply try sliding the lid off instead. That won't create any vaccuum at all.