  We show that any toroidal DM stack $X$ with finite diagonalizable inertia
possesses a maximal toroidal coarsening $X_{tcs}$ such that the morphism $X\to
X_{tcs}$ is logarithmically smooth.
  Further, we use torification results of [AT17] to construct a
destackification functor, a variant of the main result of Bergh [Ber17], on the
category of such toroidal stacks $X$. Namely, we associate to $X$ a sequence of
blowings up of toroidal stacks $\widetilde{\mathcal{F}}_X\:Y\longrightarrow X$
such that $Y_{tc}$ coincides with the usual coarse moduli space $Y_{cs}$. In
particular, this provides a toroidal resolution of the algebraic space
$X_{cs}$.
  Both $X_{tcs}$ and $\widetilde{\mathcal{F}}_X$ are functorial with respect to
strict inertia preserving morphisms $X'\to X$.
  Finally, we use coarsening morphisms to introduce a class of
non-representable birational modifications of toroidal stacks called Kummer
blowings up.
  These modifications, as well as our version of destackification, are used in
our work on functorial toroidal resolution of singularities.
