For $\overline{\mathcal{M}}_g$ (the DM compactification of the moduli of genus-$g$ curves), Chow's lemma provides a scheme $Z$ with a proper birational map $Z \to \overline{\mathcal{M}}_g$. Concretely, one can take $Z$ to be the Hilbert scheme of $n$-canonically embedded stable curves (for $n$ sufficiently large), which is a projective scheme. The map $Z \to \overline{\mathcal{M}}_g$ forgets the embedding.

This is how Mumford and Knudsen originally proved the projectivity of $\overline{M}_g$ (the coarse moduli space): $Z \to \overline{\mathcal{M}}_g \to \overline{M}_g$ with $Z$ projective and the composition proper and birational implies $\overline{M}_g$ is projective.

For $\mathcal{X} = [X/G]$ where $X$ is a proper scheme over $S$ and $G$ is a finite group, Chow's lemma for $\mathcal{X}$ can be deduced from Chow's lemma for $X$: if $X' \to X$ is a proper birational morphism with $X'$ quasi-projective, then $Z = X'$ with the composition $X' \to X \to [X/G]$ satisfies the conditions.

In fact, one can do better: the equivariant Chow's lemma provides a $G$-equivariant proper birational map $X' \to X$ with $X'$ quasi-projective and the $G$-action on $X'$ being linearized (i.e., $X'$ embeds $G$-equivariantly into some $\mathbb{P}^N$). Then $[X'/G] \to [X/G]$ is a proper birational morphism of stacks.

Chow's lemma for stacks implies the following finiteness result: if $\mathcal{X}$ is a proper DM stack over a Noetherian ring $R$ and $\mathcal{F}$ is a coherent sheaf on $\mathcal{X}$, then $H^i(\mathcal{X}, \mathcal{F})$ is a finitely generated $R$-module for all $i$.

Proof sketch: By Chow's lemma, there is a proper birational map $g : Z \to \mathcal{X}$ with $Z$ a projective scheme. The Leray spectral sequence $H^p(\mathcal{X}, R^q g_* g^*\mathcal{F}) \Rightarrow H^{p+q}(Z, g^*\mathcal{F})$ relates the cohomology of $\mathcal{X}$ to the (known-to-be-finite) cohomology of the projective scheme $Z$.

For a proper DM stack $\mathcal{X}$ over $\mathbb{C}$, the GAGA theorem for stacks (due to various authors, including Behrend, Olsson) states: $\operatorname{Coh}(\mathcal{X}) \cong \operatorname{Coh}(\mathcal{X}^{\mathrm{an}})$ (the algebraic and analytic categories of coherent sheaves are equivalent). Chow's lemma reduces this to the scheme case (Serre's original GAGA).

This implies: every analytic coherent sheaf on $\overline{\mathcal{M}}_g$ (as a complex-analytic orbifold) is algebraic.

The proper base change theorem extends to DM stacks: if $f : \mathcal{X} \to \mathcal{Y}$ is a proper morphism of DM stacks and $\mathcal{F}$ is a torsion etale sheaf on $\mathcal{X}$, then the formation of $R^i f_* \mathcal{F}$ commutes with base change. Chow's lemma reduces this to the scheme case.

Using Chow's lemma, one can extend Kodaira-type vanishing theorems to DM stacks. For a smooth proper DM stack $\mathcal{X}$ over $\mathbb{C}$ with an ample line bundle $\mathcal{L}$ (on the coarse space):

$H^i(\mathcal{X}, \omega_{\mathcal{X}} \otimes \pi^*\mathcal{L}) = 0 \quad \text{for } i > 0$

where $\pi : \mathcal{X} \to M$ is the coarse moduli map.

For $\overline{\mathcal{M}}_g$: applying this with $\mathcal{L}$ chosen appropriately on $\overline{M}_g$ gives vanishing results used to compute the Picard group of $\overline{\mathcal{M}}_g$.

Chow's lemma underpins the construction of Chow groups $A_*(\mathcal{X})$ for DM stacks (Vistoli, Edidin--Graham). The proper pushforward $g_* : A_*(Z) \to A_*(\mathcal{X})$ from a Chow's lemma presentation $Z \to \mathcal{X}$ allows one to define cycles on $\mathcal{X}$ in terms of cycles on the scheme $Z$.

The rational Chow ring $A^*(\mathcal{X})_\mathbb{Q}$ is particularly well-behaved: for $\mathcal{X} = [X/G]$ with $G$ finite, $A^*(\mathcal{X})_\mathbb{Q} \cong A^*_G(X)_\mathbb{Q}$ (equivariant Chow groups).

For a smooth proper DM stack $\mathcal{X}$ of dimension $n$, Serre duality takes the form: $H^i(\mathcal{X}, \mathcal{F}) \cong H^{n-i}(\mathcal{X}, \mathcal{F}^\vee \otimes \omega_\mathcal{X})^\vee.$

The proof uses Chow's lemma to reduce to the scheme case: take a proper birational map $g : Z \to \mathcal{X}$ with $Z$ projective, apply Serre duality on $Z$, and use the Leray spectral sequence to transfer the result back to $\mathcal{X}$.

For $\overline{\mathcal{M}}_g$: $\omega_{\overline{\mathcal{M}}_g} = \lambda_g$ (the Hodge bundle determinant, up to boundary corrections), and Serre duality relates $H^i(\overline{\mathcal{M}}_g, \lambda_g^{\otimes k})$ to $H^{3g-3-i}(\overline{\mathcal{M}}_g, \lambda_g^{\otimes(1-k)})$.

For a smooth proper DM stack $\mathcal{X}$ and a "hyperplane section" $\mathcal{H} \hookrightarrow \mathcal{X}$ (defined via the coarse space), the Lefschetz hyperplane theorem states: $H^i(\mathcal{X}, \mathbb{Q}_\ell) \xrightarrow{\sim} H^i(\mathcal{H}, \mathbb{Q}_\ell) \quad \text{for } i < \dim \mathcal{X} - 1.$

Chow's lemma allows one to reduce this to the scheme case by pulling back along $g : Z \to \mathcal{X}$.

A consequence of Chow's lemma (combined with the Totaro--Edidin--Graham embedding theorem): every smooth separated DM stack $\mathcal{X}$ of finite type with quasi-projective coarse space is a global quotient stack $[U/\mathrm{GL}_N]$ for some quasi-projective scheme $U$ and some $N$.

This is crucial for defining equivariant cohomology theories: since $\mathcal{X} = [U/\mathrm{GL}_N]$, one has $H^*(\mathcal{X}) = H^*_{\mathrm{GL}_N}(U)$, and the Borel construction gives a topological model.

For the moduli stack $\operatorname{Quot}_{E/X/S}^P$ of quotients of a coherent sheaf $E$ on $X/S$, Chow's lemma (applied to the already-known projectivity of the coarse space) confirms that the stack itself has good cohomological properties. In particular, the obstruction theory and virtual fundamental class on $\operatorname{Quot}$ are well-defined.

The same applies to the stack of stable maps $\overline{\mathcal{M}}_{g,n}(X, \beta)$: Chow's lemma for the coarse space $\overline{M}_{g,n}(X, \beta)$ (which is projective by Kontsevich) ensures that the virtual fundamental class on the stack pushes forward correctly.

In the context of the minimal model program for stacks (developed by Kresch, Abramovich, and others), Chow's lemma plays an analogous role to the classical case: it allows one to reduce questions about proper birational maps of stacks to questions about projective morphisms. This is used, for instance, in the proof that $\overline{M}_g$ is of general type for large $g$, where one needs to construct effective divisors on $\overline{\mathcal{M}}_g$ and compute their classes in the Picard group.

As a companion to Chow's lemma, there is a Nagata compactification theorem for stacks (Rydh): every separated morphism $\mathcal{X} \to S$ of finite type with $\mathcal{X}$ a DM stack admits a compactification, i.e., there exists a proper DM stack $\bar{\mathcal{X}}$ over $S$ containing $\mathcal{X}$ as a dense open substack.

Combined with Chow's lemma, this gives: every separated DM stack of finite type admits a proper birational map from a quasi-projective scheme, which is the starting point for many cohomological arguments.