  There are many classical results, related to the Denjoy--Wolff Theorem,
concerning the relationship between orbits of interior points and orbits of
boundary points under iterates of holomorphic self-maps of the unit disc. Here,
for the first time, we address such questions in the very general setting of
sequences $(F_n)$ of holomorphic maps between simply connected domains. We show
that, while some classical results can be generalised, with an interesting
dependence on the geometry of the domains, a much richer variety of behaviours
is possible. Some of our results are new even in the classical setting.
  Our methods apply in particular to non-autonomous dynamical systems, when
$(F_n)$ are forward compositions of holomorphic maps, and to the study of
wandering domains in holomorphic dynamics.
  The proofs use techniques from geometric function theory, measure theory and
ergodic theory, and the construction of examples involves a `weak independence'
version of the second Borel--Cantelli lemma and the concept from ergodic theory
of `shrinking targets'.
