Let $C$ be a connected smooth curve, $0\in C$ a closed point and $W\rightarrow C$ a family of projective schemes. Assume that the fibers $W_t$ of $W$ are smooth for all $t\neq 0$ and that $W_0=Y_1\cup Y_2$ where $Y_1, Y_2$ are smooth varieties intersecting at a smooth divisor.
Jun Li constructs a (proper DM) stack $\mathcal{M}(\mathfrak{M},\Gamma)$ of stable morphisms to the stack $\mathfrak{M}$ of expanded degenerations of $W/C$ of topological type $\Gamma$. According to lemma 3.10 in the paper "Stable morphisms to singular schemes and relative stable morphisms", there exists a surjective map $\mathcal{M}(W[n],\Gamma)^{st}\rightarrow \mathcal{M}(\mathfrak{M},\Gamma)$ for sufficiently large $n$. Here, $\mathcal{M}(W[n],\Gamma)^{st}$ is the locally closed substack of the stack of stable maps to $W[n]$ which are nondegenerate, predeformable and stable (in the sense of definition 3.1) morphisms of type $\Gamma$ to $W[n]$.
There is a $({\mathbb{C}^{*}})^n$ action on $W[n]$, which in turn induces a $({\mathbb{C}^{*}})^n$ action on $\mathcal{M}(W[n],\Gamma)^{st}$ with finite stabilizers. Also there is an induced map $\mathcal{M}(W[n],\Gamma)^{st}/({\mathbb{C}^{*}})^n\rightarrow \mathcal{M}(\mathfrak{M},\Gamma)$, which is claimed to be étale and finite.
I don't understand the last claim. Is that map representable? Why is it proper? (I understand quasifinite.) Assuming the claim, $\mathcal{M}(W[n],\Gamma)^{st}/({\mathbb{C}^{*}})^n$ has to be proper. Is it obvious that $\mathcal{M}(W[n],\Gamma)^{st}/({\mathbb{C}^{*}})^n$ is proper? ($\mathcal{M}(W[n],\Gamma)^{st}$ is only separated finite type.) An explicit description of the map would be welcome, too.
