[SidiousL]

Your intuition is correct and the question of the speed of gravitational waves is pretty complicated. The usual treatment of gravitational waves is done by linearization of some non-linear equations. This is a very good approximation for the propagation of gravitational waves since they disturb the space-time only very slightly (for example, the mirrors in the interferometry experiment at LIGO get displaced by a tenth of a nucleus of a hydrogen atom, during the passage of a gravitational wave).

In this linear approximation the gravitational waves are governed by the same wave equation as for electromagnetism (only the spin part is different since the spins of the gravitons and photons are different). Since in the linear approximation we recover the Lorentz symmetry, then the linearized wave equation has to be Lorentz invariant. Then, one can apply results from the representation theory of the Lorentz or Poincare groups; there are two major types of representations: massless and massive. They differ in striking ways when it comes to spin and when it comes to propagation, for example massless particles travel at the speed of light. If you want to have a massive graviton then you need to get the mass somehow from your theory. Einstein's theory predicts a massless graviton, which by the argument above has to travel at the speed of light. We still don't know experimentally if the graviton is massless (but last time I looked at the Particle Data Book there was an upper bound on the mass which was very small).

Now, coming back to why your question is complicated. Remember that in Einstein's theory space-time itself is dynamical. Now suppose you follow the propagation of a gravitational wave. Since this takes some time, we need to take into account the fact that the shape of the space-time itself has changed in the meantime. In such dynamical situations it becomes complicated to even define what the speed of propagation between two points is. One way this becomes important is in cosmological situations. For example, the expansion of the universe stretches the distances and this allows us for example to see further that the distance you obtain multiplying the speed of light by the age of the universe. This being said, you can define the speed locally by studying propagation for very short distances and times and this will be a constant. However, you need to remember that the global situation is more complicated.

[saganus]

This is what I was thinking. I.e. that it would be very complicated. I wasn't even sure if it made sense to ask such a question.

I have so much to read now...