Given algebraic spaces $X$, $Y$, $Z$ with a finite morphism $Y \rightarrow X$ and a closed immersion $Y \hookrightarrow Z$, the pushout $P \cong X \amalg_Y Z$ exists as an algebraic space (cf. Temkin and Tyomkin - Ferrand pushouts for algebraic spaces, Theorem 6.2.1 (ii),(b)).
Does this still hold if everywhere above we replace "algebraic space" by "algebraic stack" (or "DM stack")?
Yes, this is exactly Theorem A.4 in my old preprint Compactification of tame Deligne–Mumford stacks which is long overdue to appear on the arXiv. The proof is rather terse but fairly standard (compare with Appendix A of Jack Hall's Openness of versality via coherent functors). A more detailed proof will also appear in the upcoming paper:
Artin algebraization for pairs with applications to the local structure of stacks and Ferrand pushouts (now on arXiv:2205.08623)
which is joint with Jarod Alper, Jack Hall and Daniel Halpern-Leistner. There we prove the existence of pushouts of affine morphisms and closed immersions in the category of (quasi-separated) algebraic stacks.
PS. At least some authors call pushouts of finite morphisms and closed immersions for "pinchings". Ferrand considered the more general case of affine morphisms, hence the name Ferrand pushouts.
