  Let $G$ and $\Omega$ be two planar domains. We give necessary and sufficient
conditions on a sequence $(\phi_n)$ of eventually injective holomorphic
mappings from $G$ to $\Omega$ for the existence of a function $f\in H(\Omega)$
whose orbit under the composition by $(\phi_n)$ is dense in $H(G)$. This
extends a result of the same nature obtained by Grosse-Erdmann and Mortini when
$G=\Omega$. An interconnexion between the topological properties of $G$ and
$\Omega$ appears. Further, in order to exhibit in a natural way holomorphic
functions with wild boundary behaviour on planar domains, we study a certain
type of universality for sequences of continuous mappings from a union of
Jordan curves to a domain.
