For a suitable Markov Process $(X_t)_{t\in [0,\infty)}$ with state space $E$ we may have the Strong Markov Property such that for a measurable and postive function $\phi:E^{[0,\infty)} \rightarrow \Bbb R^+$ and a a.s. finite stopping time $T$ it holds: $$\Bbb E _ x [\phi((X_{t+T})_{t\in [0,\infty)})| \mathcal{F}_T] = \Bbb E _ {X_T} [\phi((X_{t})_{t\in [0,\infty)})]$$ In my opinion for a measurable function $g:E\rightarrow \Bbb R ^+$ should hold something like $$\Bbb E _ x [g(X_t)\Bbb 1_{\{T \leq t\}}|\mathcal{F}_T] (\omega) = (\Bbb 1_{\{T \leq t\}} \Bbb E _ {X_T} [g(X_{t-s})]_{s=T})(\omega) := \Bbb 1_{\{T \leq t\}}(\omega) \, \Bbb E _ {X_T(\omega)} [g(X_{t-T(\omega)})]$$ but I am not quite sure. I cannot find a proof for it, though i think if this is true there should be a way to prove it with the strong Markov property.
May one need additional conditions so that this holds.