If $Y$ is a Markov process, how can we calculate $\operatorname P\left[Y_{s+t-\sigma};\sigma\le s+t<\tau\in B\mid\mathcal F_s\right]$?

Question

Let

• $(E,\mathcal E)$ be a measurable space;
• $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$;
• $(\Omega,\mathcal A,\operatorname P)$ be a probability space;
• $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$;
• $(Y_t)_{t\ge0}$ be an $(E,\mathcal E)$-valued process on $(\Omega,\mathcal A)$ with $$\operatorname P\left[Y_{s+t}\in B\mid\mathcal F_s\right]=\kappa_t(Y_s,B)\tag1$$ for all $B\in\mathcal E$ and $s,t\ge0$.

Let $\sigma,\tau$ be $(\mathcal F_t)_{t\ge0}$-stopping times on $(\Omega,\mathcal A)$. How can we calculate $\operatorname P\left[Y_{s+t-\sigma}\in B;\sigma\le s+t<\tau\mid\mathcal F_s\right]$?

Assuming that $(Y_t)_{t\ge0}$ is strongly $(\mathcal F_t)_{t\ge0}$-Markovian, i.e. $$\operatorname P\left[X_{\zeta+t}\in B;\zeta<\infty\mid\mathcal F_\zeta\right]=\kappa_t(X_\zeta,B)\;\;\;\text{on }\{\zeta<\infty\}\tag2$$ for all $B\in\mathcal E$ and $t\ge0$. Can we use this to find an expression?