If $(Y_r)_{r∈[0,\:t]}$ is a càdlàg process, is there some relation between the distribution of $(Y_r)_{r∈[0,\:s]}$ and $(Y_r)_{r∈[0,\:t]}$ for $s≤t$?

Question

Let $t>0$, $E$ be a separable metric space and $$D([0,t],E):=\left\{x:[0,t]\to E\mid x\text{ is càdlàg}\right\}$$ be equipped with the Skorohod topology.

Now, let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $Y:\Omega\to D([0,t],E)$ be a $(\mathcal A,\mathcal B(D([0,t],E)))$-measurable, $s\in(0,t]$ and $X:=\left.Y\right|_{[0,\:s]}$.

Question: Is there a relation between $\operatorname P_\ast X$ and $\operatorname P_\ast Y$?

I could imagine that there is some relation in terms of the trace $\sigma$-algebra, but I wasn't able to figure out details.