I put my extracted and partially capped frames above the inner cover for the bees to take down the honey. But I
also leave them on until after the fall flow of aster and goldenrod. So the bees first take down the honey then
put any excess above the inner cover in the supers. I usually get a super or two od fall honey.
So the answer is yes, they will take the honey down, but, if the lower boxes are full, they will put honey abover
the inner cover too. Based on what you said, they should clean them out nicely.