  We study integral kernels of strongly continuous semigroups on Lebesgue
spaces over metric measure spaces. Based on semigroup smoothing properties and
abstract Morrey-type inequalities, we give sufficient conditions for H\"older
or Lipschitz continuity of the kernels. We apply our results to
(pseudo)differential operators on domains and quantum graphs, to Laplacians on
a class of fractals including the Sierpi\'nski gasket, and to structurally
damped wave equations. An extension to non-autonomous problems is also
discussed.
