This example is about moduli of weighted pointed stable curves. In this case the knowledge of the boundary objects is fundamental to work out the birational geometry (for instance in relation to the log minimal model program) of the spaces themselves.

Let $S$ be a Noetherian scheme and $g,n$ two non-negative integers. A family of nodal curves of genus $g$ with $n$ marked points over $S$ consists of a flat proper morphism $\pi:C\rightarrow S$ whose geometric fibers are nodal connected curves of arithmetic genus $g$, and sections $s_{1},...,s_{n}$ of $\pi$. A collection of input data $(g,A) := (g, a_{1},...,a_{n})$ consists of an integer $g\geq 0$ and the weight data: an element $(a_{1},...,a_{n})\in\mathbb{Q}^{n}$ such that $0<a_{i}\leq 1$ for $i = 1,...,n$, and
$$2g-2 + \sum_{i = 1}^{n}a_{i} > 0.$$
A family of nodal curves with marked points $\pi:(C,s_{1},...,s_{n})\rightarrow S$ is stable of type $(g,A)$ if

- the sections $s_{1},...,s_{n}$ lie in the smooth locus of $\pi$, and for any subset $\{s_{i_{1}}, . . . , s_{i_{r}}\}$ with non-empty intersection we have $a_{i_{1}} +...+ a_{i_{r}} \leq 1$,
- $K_{\pi}+\sum_{i=1}^{n}a_{i}s_{i}$ is $\pi$-relatively ample.

Now, given a collection $(g,A)$ of input data, there exists a connected Deligne-Mumford stack $\overline{\mathcal{M}}_{g,A[n]}$, smooth and proper over $\mathbb{Z}$, representing the moduli problem of pointed stable curves of type $(g,A)$. The corresponding coarse moduli scheme $\overline{M}_{g,A[n]}$ is projective over $\mathbb{Z}$.

B. Hassett, *"Moduli spaces of weighted pointed stable curves"*, Advances in Mathematics 173 (2003), Issue 2, 316-352.

Fixed $g,n$, consider two collections of weight data $A[n],B[n]$ such that $a_i\geq b_i$ for any $i = 1,...,n$. Then there exists a birational \textit{reduction morphism}
$$\rho_{B[n],A[n]}:\overline{M}_{g,A[n]}\rightarrow\overline{M}_{g,B[n]}$$
associating to a curve $[C,s_1,...,s_n]\in\overline{M}_{g,A[n]}$ the curve $\rho_{B[n],A[n]}([C,s_1,...,s_n])$ obtained by collapsing components of $C$ along which $K_C+b_1s_1+...+b_ns_n$ fails to be ample.

The reduction morphisms are defined in terms of the parametrized objects. Some of the spaces $\overline{M}_{0,A[n]}$ endowed with these reduction morphisms appear as intermediate steps of Kapranov's blow-up construction of $\overline{M}_{0,n}$,

M. Kapranov, *"Veronese curves and Grothendieck-Knudsen moduli spaces $\overline{M}_{0,n}$"*, Jour. Alg. Geom. 2 (1993), 239-262.

In higher genus $\overline{M}_{g,A[n]}$ may be related to the log minimal model program on $\overline{M}_{g,n}$,

H. Moon, *"Log canonical models for $\overline{M}_{g,n}$"*, https://archive.org/details/arxiv-1111.5354.