The classic application is Deligne & Mumford's [paper][1] proving the irreducibility of the coarse moduli scheme $\overline{M}_g$ of stable genus g curves over any algebraically closed field.  They proved irreducibility first for the moduli stack $\overline{\mathcal{M}}_g$ of curves, and then inferred from this result the irreducibility of the scheme.

  [1]: http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1969__36_/PMIHES_1969__36__75_0/PMIHES_1969__36__75_0.pdf