Does gravity propagate? Gravity is sometimes described as a curvature in space-time. Due to relativity, doesn't this imply that gravity doesn't propagate?

A thought experiment. If you are moving toward a black hole, this could also be considered as if the black hole is moving toward you. If a black hole was moving toward you at the speed of light, and gravity propagates at the speed of light, you would experience something similar to a supersonic blast, except its a gravity blast, and it would allow you to remain unaffected until you arrived at the very centre of the black hole?

If you were able to turn around, just before you arrive at the center, and travel away from the black hole at the speed of light you would never notice it, even though you would be travelling at the same velocity as the black hole in the same direction?
Black holes distort the geometry of spacetime. So you need to take care about that.

Changes of gravity propagate with the speed of light. That's thought to occur for accelerated mass, e.g. a binary of black holes.

A second fundamental point: Speeds don't add up in an additive way, but in a subadditive way: speed of light + speed of light = speed of light.

If you or the black hole travel with the speed of light, you resp. the black hole needs to be massless at rest, otherwise your relativistic mass, or that of the black hole would be infinite.

Photons are massless at rest, so you're actually talking of light, or something which behaves rather similar to light in many aspects.

So at the end you're talking of light directed to a black hole and reflected by a mirror inside the black hole.

The light can get inside the black hole, but it cannot leave the hole after being reflected from the mirror due to the high curvature of spacetime below the event horizon.

If you are travelling towards a binary of black holes, you get blue-shifted changes of gravity.