Disintegrate or just stop and drop out of warp just like the ship does when they disengage warp normally. The ship doesn’t travel faster than the speed of light. It just warps space around the ship. That’s why they don’t have relativistic speeds like the warping of time like time dilation on the ship doesn’t happen.

I would think that any object (including the ship) is traveling at a sub-light speed within the warp bubble and therefore would only keep that same velocity when (catastrophically) exiting the warp bubble. Unless by exiting the warp bubble in an uncontrolled manner creates some other force which slows the object somehow.

My understanding is that the warp bubble is moving space around the object (including the ship) rather than accelerating the object to FTL (faster-than-light) speeds, thus we really only have to consider the relative velocities within the warp bubble.

Edit to add: Oh, also, I should add that (IMO) the object cannot continue to travel at FTL speed since it has no warp drive of its own to maintain the warp field.

I think we’ve seen enough to safely say it would fall out of FTL in fairly short order.

I don’t think we’ve seen enough to say exactly what its sublight velocity would be - no examples are coming to mind.

Edit: as is often the case when the warp drive comes up, its worth pointing out that ships at warp pretty definitely have momentum/inertia.

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.