  Our study is focused on the dynamics of weighted composition operators
defined on a locally convex space $E\hookrightarrow (C(X),\tau_p)$ with $X$
being a topological Hausdorff space containing at least two different points
and such that the evaluations $\{\delta_x:\ x\in X\}$ are linearly independent
in $E'$. We prove, when $X$ is compact and $E$ is a Banach space containing a
nowhere vanishing function, that a weighted composition operator
$C_{\varphi,\omega}$ is never weakly supercyclic on $E$. We also prove that if
the symbol $\varphi$ lies in the unit ball of $A(\mathbb{D})$, then every
weighted composition operator can never be $\tau_p$-supercyclic neither on
$C(\mathbb{D})$ nor on the disc algebra $A(\mathbb{D})$. Finally, we obtain
Ansari-Bourdon type results and conditions on the spectrum for arbitrary weakly
supercyclic operators, and we provide necessary conditions for a composition
operator to be weakly supercyclic on the space of holomorphic functions defined
in non necessarily simply connected planar domains. As a consequence, we show
that no composition operator can be weakly supercyclic neither on the space of
holomorphic functions on the punctured disc nor in the punctured plane.
