Let $E$ be a locally compact separable$^1$ metric space, $D([0,\infty),E)$ denote the Skorohod space, $C_0(E)$ denote the space of continuous functions $E\to\mathbb R$ vanishing at infinity and $E^\ast:=E\uplus\left\{\infty\right\}$ denote the Alexandroff one-point compactification.
Moreover, let $\mathcal M_1(D([0,\infty),E))$ denote the space of probability measures on $\mathcal B(D([0,\infty),E))$ equipped with the topology of weak convergence$^1$.
Now let $X^n$ be a $D([0,\infty),E)$-valued random variable on a probability space $(\Omega,\mathcal A,\operatorname P)$ for $n\in\mathbb N$.
I want to show that if $\left(\left(f\circ X^n\right)_\ast\operatorname P\right)_{n\in\mathbb N}\subseteq\mathcal M_1(D([0,\infty),\mathbb R))$ is relatively compact for all $f\in C_0(E)$, then $\left(X^n_\ast\operatorname P\right)_{n\in\mathbb N}\subseteq\mathcal M_1(D([0,\infty),E^\ast))$ is relatively compact.
This is claimed and proved in Corollary 9.3 of Chapter 3 in the book of Ethier and Kurtz. However, I don't understand their proof and hope there is an easier proof available which doesn't rely on the more general Theorem 9.1 (of the same chapter).
It's clear to me that if $f^\ast\in C(E^\ast)$, then $$f:=\left.f^\ast\right|_E-f^\ast(\infty)\in C_0(E)\tag1$$ and hence, by assumption, $\left(\left(f\circ X^n\right)_\ast\operatorname P\right)_{n\in\mathbb N}\subseteq\mathcal M_1(D([0,\infty),\mathbb R))$ is relatively compact. However, I don't understand how they conclude that $\left(\left(f\circ X^n\right)_\ast\operatorname P\right)_{n\in\mathbb N}\subseteq\mathcal M_1(D([0,\infty),\mathbb R))$ is actually relatively compact for all $f\in C(E^\ast)$ (not only those which arise by $(1)$).
And even when this has been shown, how do we conclude the claim? As I said before, Theorem 9.1 is quite general. For example, since $E^\ast$ is compact, the "compact containment condition" (9.1) should be trivially satisfied. So, how can we proceed to prove the claim?
$^1$ I guess it's crucial to assume that $E$ (and hence $D([0,\infty),E)$) is separable, since then $\mathcal M_1(D([0,\infty),E))$ is metrizable which in turn implies that relative sequential compactness and relative compactness are equivalent. Maybe someone can elaborate on that.