Invariance of plurigenera and good minimal models

Nakayama showed that deformation invariance of plurigenera for smooth complex varieties follows from the MMP and Abundance Conjectures. We generalize his result to families of singular pairs over DVRs of positive or mixed characteristic. As an application we show invariance of plurigenera for deformations of good minimal models of general type varieties whose canonical model has rational singularities as well as boundedness of Gorenstein canonical models of dimension three and fixed volume over an algebraically closed field of characteristic $p>5$.