  A well-known problem in holomorphic dynamics is to obtain Denjoy--Wolff-type
results for compositions of self-maps of the unit disc. Here, we tackle the
particular case of inner functions: if $f_n:\mathbb{D}\to\mathbb{D}$ are inner
functions fixing the origin, we show that a limit function of
$f_n\circ\cdots\circ f_1$ is either constant or an inner function. For the
special case of Blaschke products, we prove a similar result and show,
furthermore, that imposing certain conditions on the speed of convergence
guarantees $L^1$ convergence of the boundary extensions. We give a
counterexample showing that, without these extra conditions, the boundary
extensions may diverge at all points of $\partial\mathbb{D}$.
