What would happen if you were pushed (i.e. transported) by Kuma's Devil Fruit powers while you were trapped inside Doflamingo's birdcage? Will the victim get wounded or injured by the slicing strings of the Birdcage when transported inside Kuma's Paw-like bubble? Or will the Birdcage give way for the Paw-like bubble to successfully escape?
Since there are no facts that supports any of the 2 scenarios, I will say that the Bridcage strings would tear the bubble apart since a bunch of DF-users tried unsucessfully to stop the cage strings from moving.
Following "one piece logic" I am inclined to say that Kuma's paw-bubble isn't invincible (since it is made out of "air-like" shield) as it would one of the best DF there is if the user can make objects invulnerable. Therefore I say that the Kuma's bubble wouldn't pass through Doflamingo's Birdcage
Difficult to say, nothing is verified but I would guess that the birdcage would work like the ground and just stop the travel of the bubble and the person inside would land on the birdcage (and fall if it's on the inside).
Why I say that? It's because the bubble is like a block (can physically transport a person) So, for me, it should work the same way for the string and wouldn't pass through it.