  Complementing earlier results on dynamics of unilateral weighted shifts, we
obtain a sufficient (but not necessary, with supporting examples) condition for
hypercyclicity, mixing and chaos for $M_z^*$, the adjoint of $M_z$, on
vector-valued analytic reproducing kernel Hilbert spaces $\mathcal{H}$ in terms
of the derivatives of kernel functions on the open unit disc $\mathbb{D}$ in
$\mathbb{C}$. Here $M_z$ denotes the multiplication operator by the coordinate
function $z$, that is \[ (M_z f) (w) = w f(w), \] for all $f \in \mathcal{H}$
and $w \in \mathbb{D}$. We analyze the special case of quasi-scalar reproducing
kernel Hilbert spaces. We also present a complete characterization of
hypercyclicity of $M_z^*$ on tridiagonal reproducing kernel Hilbert spaces and
some special classes of vector-valued analytic reproducing kernel Hilbert
spaces.
