Representation stability for the Kontsevich space of stable maps

For a fixed algebraic variety $X$, curve class $\alpha \in N_1(X)$, and genus $g\in \mathbb{N}$, we consider the sequence of $S_n$ representations obtained from the homology of the Kontsevich space of stable maps to $X$, $\bar M_{g,n}(X,\alpha)$. Using the category of finite sets and surjections, we prove a representation stability theorem that governs the behavior of this sequence of representations for $n$ sufficiently large.