  We study topological transitivity/hypercyclicity and topological (weak)
mixing for weighted composition operators on locally convex spaces of
scalar-valued functions which are defined by local properties. As main
application of our general approach we characterize these dynamical properties
for weighted composition operators on spaces of ultradifferentiable functions,
both of Beurling and Roumieu type, and on spaces of zero solutions of elliptic
partial differential equations. Special attention is given to eigenspaces of
the Laplace operator and the Cauchy-Riemann operator, respectively. Moreover,
we show that our abstract approach unifies existing results which characterize
hypercyclicity, resp. topological mixing, of (weighted) composition operators
on the space of holomorphic functions on a simply connected domain in the
complex plane, on the space of smooth functions on an open subset of
$\mathbb{R}^d$, as well as results characterizing topological transitiviy of
such operators on the space of real analytic functions on an open subset of
$\mathbb{R}^d$.
