  We prove sharp results about recurrent behaviour of orbits of forward
compositions of inner functions, inspired by fundamental results about iterates
of inner functions, and give examples to illustrate behaviours that cannot
occur in the simpler case of iteration.
  A result of Fern\'andez, Meli\'an and Pestana gives a precise version of the
classical Poincar\'e recurrence theorem for iterates of the boundary extension
of an inner function that fixes~0. We generalise this to forward composition
sequences $F_n=f_n\circ \dots\circ f_1,$ $n\in \mathbb{N},$ where $f_n$ are
inner functions that fix~0, giving conditions on the contraction of $(F_n)$ so
that the radial boundary extension $F_n$ hits any shrinking target of arcs
$(I_n)$ of a given size.
  Next, Aaronson, and also Doering and Ma\~n\'e, gave a remarkable dichotomy
for iterates of any inner function, showing that the behaviour of the boundary
extension is of two entirely different types, depending on the size of the
sequence $(|f^n(0)|)$. In earlier work, we showed that one part of this
dichotomy holds in the non-autonomous setting of forward compositions.
  It turns out that this dichotomy is closely related to the result of
Fern\'andez, Meli\'an and Pestana, and here we show that a version of the
second part of the dichotomy holds in the non-autonomous setting provided we
impose a condition on the contraction of $(F_n)$ in relation to the size of the
sequence $(|F_n(0)|)$. The techniques we use include a strong version of the
second Borel--Cantelli lemma and strong mixing results of Pommerenke for
contracting sequences of inner functions. We give examples to show that the
contraction conditions that we need to impose in the non-autonomous setting are
best possible.
