We study $\mathbb{S}_n$-equivariant motivic invariants of the moduli space $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$ of degree-$d$ maps from $n$-pointed curves of genus $g$ to $\mathbb{P}^r$. In particular, we obtain formulas for the Serre characteristic, which specializes to the Hodge--Deligne polynomial. Fixing $g, r \geq 1$, we prove that an explicit invertible transform of the generating function for the Serre characteristics is rational. We use our formula to prove a stability result for the weight-graded compactly-supported Euler characteristics of $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$ as $d \to \infty$. In genus one and two, we reduce the calculation of the Serre characteristic of $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$ to those of the moduli spaces $\mathcal{M}_{g, n}$ of $n$-pointed curves. Formulas for the latter follow from work of Getzler and Petersen, so our formula in particular determines the Serre characteristic of $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$ for arbitrary $n$, $r$, and $d$ when $g = 1$ and $g = 2$.