In classical theories of gravitation, the changes in a gravitational field propagate. A change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational field which it produces. In a more physically correct sense, the "speed of gravity" refers to the speed of a gravitational wave, which is the same speed as the speed of light (c).
The speed of gravitational waves in the general theory of relativity is equal to the speed of light in vacuum, c. Within the theory of special relativity, the constant c is not exclusively about light; instead it is the highest possible speed for any interaction in nature. Formally, c is a conversion factor for changing the unit of time to the unit of space. This makes it the only speed which does not depend either on the motion of an observer or a source of light and/or gravity. Thus, the speed of "light" is also the speed of gravitational waves and any other massless particle. Such particles include the gluon (carrier of the strong force), the photons that make up light, and the theoretical gravitons which make up the associated field particles of gravity (however a theory of the graviton requires a theory of quantum gravity).
The speed of physical changes in a gravitational or electromagnetic field should not be confused with "changes" in the behavior of static fields that are due to pure observer-effects. These changes in direction of a static field, because of relativistic considerations, are the same for an observer when a distant charge is moving, as when an observer (instead) decides to move with respect to a distant charge. Thus, constant motion of an observer with regard to a static charge and its extended static field (either a gravitational or electric field) does not change the field. For static fields, such as the electrostatic field connected with electric charge, or the gravitational field connected to a massive object, the field extends to infinity, and does not propagate. Motion of an observer does not cause the direction of such a field to change, and by symmetrical considerations, changing the observer frame so that the charge appears to be moving at a constant rate, also does not cause the direction of its field to change, but requires that it continue to "point" in the direction of the charge, at all distances from the charge.