# If $X^n$ is a sequence of càdlàg processes whose FDDs converge to a continous process $X$, does $X^n$ converge to $X$ in the Skorohod topology?

Let $$(\Omega,\mathcal A,\operatorname P)$$ be a probability space, $$E$$ be a complete locally compact separable metric space, $$(X^n_t)_{t\ge0}$$ be an $$E$$-valued càdlàg process on $$(\Omega,\mathcal A,\operatorname P)$$ for $$n\in\mathbb N$$ and $$(X_t)_{t\ge0}$$ be an $$E$$-valued continuous process on $$(\Omega,\mathcal A,\operatorname P)$$ with $$\left(X^n_{t_0},\ldots,X^n_{t_k}\right)\xrightarrow{n\to\infty}\left(X_{t_0},\ldots,X_{t_k}\right)\tag1$$ in distribution for all $$k\in\mathbb N_0$$ and $$0=t_0<\cdots.

Are we able to conclude $$X^n\xrightarrow{n\to\infty}X\tag2$$ in distribution (with respect to the Skorohod topology)?

I know that if $$S$$ is a Polish space, then a sequence $$(\mu_n)_{n\in\mathbb N}$$ of probability measures on $$(S,\mathcal B(S))$$ converges weakly to a probability measure $$\mu$$ on $$(S,\mathcal B(S))$$ if and only if $$(\mu_n)_{n\in\mathbb N}$$ is tight and there is a separating family $$\mathcal C$$ of bounded continuous functions $$S\to\mathbb R$$ such that $$\int f\:{\rm d}\mu=\lim_{n\to\infty}\int f\:{\rm d}\mu_n\;\;\;\text{for all }f\in\mathcal C\tag3.$$

In the context of the question, $$S$$ is the space $$C([0,\infty),E)$$ of càdlàg functions $$[0,\infty)\to E$$ equipped with the Skorohod metric. My hope is that we somehow can benefit from the fact that $$X$$ is not only càdlàg, but continuous.

Remark: I don't think that it's of any use, but let me remark that in my application $$X^n$$ and $$X$$ are Markov processes. Their transition semigroups, say $$(\kappa_t)_{t\ge0}$$ and $$(\kappa^n_t)_{t\ge0}$$, induce strongly continuous contraction semigroups, say $$(T_n(t))_{t\ge0}$$ and $$(T(t))_{t\ge0}$$, on the space $$C_0(E)$$ of continuous functions $$E\to\mathbb R$$ vanishing at infinity equipped with the supremum norm. Each $$(T_n(t))_{t\ge0}$$ is even uniformly continuous. Moreover, $$T_n(t)\xrightarrow{n\to\infty}T(t)$$ with respect to the strong operator topology uniformly for $$t$$ in a bounded interval. If any of these properties are useful, feel free to use them.