This paper seems to be relevant to your question: jstor.org/stable/27642669

@ThomasKojar $X$ is the Doob's $h$-transform (not with that $h$, though) of the Brownian motion (it's a 2-dimensional BM conditioned on not touching a bounded domain), and $h$ is some complicated thingy which involves the expected value of some function on the boundary of that domain wrt the entrance measure from $x$ there (by the BM). So I was a bit in doubt how to differentiate it correctly... On the other hand, that equality ${\bf E}_x (...) = h(x)$ is easy to obtain. But I've already figured out that one can insert $t\wedge \tau_r$ there instead of just $\tau_r$, thus solving my problem.

@ThomasKojar it's fine to assume some regularity for $h$ (maybe it even has to be "nice" if $X$ is a "good" diffusion --- in the example I have in mind $f$ is an analytic function). Btw, I think I've already figured out how to circumvent my specific issue; but nevertheless I'm curious how can one pass from a "sequence of stopping times"-statement to a "fixed $t$"-statement.