  We study power boundedness and related properties such as mean ergodicity for
(weighted) composition operators on function spaces defined by local
properties. As a main application of our general approach we characterize when
(weighted) composition operators are power bounded, topologizable, and
(uniformly) mean ergodic on kernels of certain linear partial differential
operators including elliptic operators as well as non-degenrate parabolic
operators. Moreover, under mild assumptions on the weight and the symbol we
give a characterisation of those weighted composition operators on the
Fr\'echet space of continuous functions on a locally compact, $\sigma$-compact,
non-compact Hausdorff space which are generators of strongly continuous
semigroups on these spaces.
