Configurations of points on degenerate varieties and properness of moduli spaces

We generalize Kim and Sato's construction of the relative configuration space $X_{D}^{[n]}$ to the case where $X$ is an algebraic stack; and construct an analogous projective moduli space $W_{\pi}^{[n]}$ for a degeneration $\pi\colon W\to B$. We construct $X_{D}^{[n]}$ and $W_{\pi}^{[n]}$ and prove their properness using a universal construction of expanded degenerations and pairs introduced in our paper with Cadman and Wise. We use these spaces to re-prove the properness of Jun Li’s moduli of relative and degenerate stable maps, and generalize this properness result to the case of target stacks.