  We study the strong continuity of weighted composition semigroups of the form
$T_tf=\varphi_t'\left(f\circ\varphi_t\right)$ in several spaces of analytic
functions. First we give a general result on separable spaces and use it to
prove that these semigroups are always strongly continuous in the Hardy and
Bergman spaces. Then we focus on two non-separable family of spaces, the mixed
norm and the weighted Banach spaces. We characterize the maximal subspace in
which a semigroup of analytic functions induces a strongly continuous semigroup
of weighted composition operators depending on its Denjoy-Wolff point, via the
study of an integral-type operator.
