  We introduce a lifting property for local cohomology, which leads to a
unified treatment of the dualizing complex for flat morphisms with
semi-log-canonical, Du Bois or F-pure fibers. As a consequence we obtain that,
in all 3 cases, the cohomology sheaves of the relative dualizing complex are
flat and commute with base change. We also derive several consequences for
deformations of semi-log-canonical, Du Bois and F-pure singularities.
