Let G and Ω be two planar domains. We give necessary and sufficient conditions on a sequence (φ n) of eventually injective holomorphic mappings from G to Ω for the existence of a function f ∈ H(Ω) whose orbit under the composition by (φ n) is dense in H(G). This extends a result of the same nature obtained by Grosse-Erdmann and Mortini when G = Ω. An interconnexion between the topological properties of G and Ω 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.