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In Newtonian physics, changes in gravity propagate instantly. In general and special relativity, gravity propagates at the speed of light, $c$. From reading answers to questions about gravity on this site, the consensus seems to be that the real world fits the relativity answer, $c$.
Numerous experiments have measured the speed of light in various media with increasing precision. Have there been any experimental measurements of the speed of gravity with generally accepted results?
If so, how did they work? Or if not, how might such a measurement be performed and why hasn't it been done yet?