Is it possible to have the analogous version of the Leibniz rule for the path integral for path integral? Something like $$\frac{\delta}{\delta\chi}\int_{\phi|_{x=x'}= \chi} [\mathcal{D}\phi]e^{-S[\phi]} \sim e^{-S[\chi]}~? $$

I have just come across a paper which involves a study about the speed of gravity. This has rocked me a bit, since I thought it was well known gravitational fields travel at the speed of light, in fact many entries on this website confirm it, e.g. How fast does gravity propagate? & "Speed" of Gra...