I'm trying to understand how we get the Markov property $$\mathbb{E}^\mu[\eta\circ\theta_t|\mathscr{G}^\circ_t] = \mathbb{E}^{Y_t}\eta\qquad \mathbf{P}^\mu\text{-a.s.}$$ in Section III.7 of Rogers and Williams Diffusions, Markov Processes and Martingales.
Here $\Omega$ is the set of paths from $[0,\infty)$ to $E_\partial$, the compactification of a locally compact, second countable Hausdorff space, $Y_t(\omega) = \omega(t)$ and $\eta\in \text{b}\mathscr{E}_\partial$ is bounded and measurable.
I know the right-hand side is $\mathscr{G}^\circ_t$ measurable, so I can use the monotone class techniques to prove the general case from a simpler case where I integrate both over the set $A = Y_r^{-1}(C)$ and $\eta = 1_\Lambda$ for $\Lambda = Y_s^{-1}(B)$ for Borel sets $B$ and $C$ and $0\le r\le t$ and $s\ge 0$.
I am stuck trying to compute $$\int_A P_s^{+\partial}(Y_t(\omega);B)\,d\mathbf{P}^\mu(\omega).$$
I'm pretty sure this is $\mathbf{P}^\mu(Y(r)\in A;Y(t+s)\in B)$, because we should have $$\int_A\mathbb{E}^\mu[\eta\circ\theta_t|\mathscr{G}^\circ_t]\,d\mathbf{P}^\mu = \int_A 1_{\theta^{-1}_t\Lambda}\,d\mathbf{P}^\mu = \mathbf{P}^\mu(A;Y(t+s)\in B).$$
I don't know how to prove this statement rigorously. Am I missing something obvious or is there something non-trivial I have to prove?