In ENT episode “Divergence” parts go flying off the ship during warp 5. What would happen to something at that speed without the protection of the field? Would it stay in high speed motion until interacted upon by something else (meteoroids/dust/gravitational fields)? Or would it disintegrate under this kind of speed?
In the “real” world, Alcubierre drives have really interesting (read “devastating”) affects on random matter interacting with the warp bubble. The bubble compresses matter in the front and creates micro singularities (which don’t necessarily go away when you drop the bubble).
In ST, I’m sure the debris does whatever the writers decide it does. I have no trouble imagining a DS9 episode in which the station gets pelted by warp velocity debris.