I understand that light cannot escape from inside of an event horizon because the spacetime curvature is too warped for photons to escape. On the other hand, gravitational waves are ripples of spacetime curvature itself (as opposed to light being IN the spacetime), so if there is some internal dynamical structures inside of blackholes (forget about no hair theorem for the moment) that produce gravitational waves, can those gravitational waves escape from inside of event horizon?
First, if you accept that gravitational waves can't travel at fast than the speed of light in regular space, then you can move to the inside of a black hole and then imagine letting the light and the gravitational wave race each other as you fall freely.
As you fall freely then over a short time interval and a short distance everything looks normal to you, you don't notice the event horizon, but you'd say that the wave can't outrun the light beam.
If the light beam can't be overtaken by the wave and the light beam can't escape, the wave can't escape either.
That's it, gravitational waves can't escape the event horizon unless they can outrun light itself in normal circumstances.
The answer to your question is that nothing can travel faster than light, and light can't escape through the event horizon. Therefore gravitational waves can't escape either. I give an algebraic proof that light can't escape in my answer to Why is a black hole black?, and a more visual proof in my answer to Would the inside of a black hole be like a giant mirror? (I'm linking my own answers because I remember them - there must be many similar answers on this site).