How can we simplify $\int_0^t\left.\frac{\rm d}{{\rm d}r}f(r,s)\:\right|_{r=s}\:{\rm d}s$ with $f(r,s)=\text E[g(X_r)h(X_{r+t-s})]$?

Question

I want to simplify an integral of the form $$\int_0^t\left.\frac{\rm d}{{\rm d}r}f(r,s)\right|_{r=s}\:{\rm d}s\tag1.$$ Anything I can generally do here? In my case, $$f(r,s)=\operatorname E\left[g(X_r)h(X_{r+t-s})\right],$$ where $X$ is a Markov process. But I don't see how this helps here.