The speeds are certainly established by theory, though. And the theory has survived every experimental test thrown at it, to date.
For instance, if Maxwell's equations were wrong, we'd start to see disagreement with experiment, even if that experiment wasn't directly designed to measure some sort of "speed".
Maxwell's equations certainly give us a good reason to expect that electromagnetism, in general, travels at 'c' in the general sense that if you change something "here", it won't have any effect "there" until after a delay of at least c/distance.
Some care does need to be taken as to what means by speed. Specifically, one has to use the above defintion, and not try and guess the speed from the direction of the coulomb force, a common sorce of confusion that is also often repeated in "speed of gravity" threads.
GR is no different as far as the theoretical aspects go. (However, we don't have any direct measurements of the speed or even the existence of gravity waves, while of course we do have direct observations of light).
The equations are a lot messier than Maxwell's equation, but there is proof that GR is a well posed initial value problem, which implies that the "fields" propagate at less than 'c'. (You can regard the "fields" as changes in the metric, which will also change the Christoffel symbols and the curvature tensor).
The details of the proof that GR is a well posed initial value problem are rather complicated and I'm not especially familiar with them, but you can find the proof in Wald, "General Relativity". I've written a little about this in the past, as to what it means to be a well-posed initial value problem and what this implies about propagation speed.