  The Denjoy-Wolff theorem is a foundational result in complex dynamics, which
describes the dynamical behaviour of the sequence of iterates of a holomorphic
self-map $f$ of the unit disc $\mathbb{D}$. Far less well understood are
nonautonomous dynamical systems $F_n=f_n\circ f_{n-1} \circ \dots \circ f_1$
and $G_n=g_1\circ g_{2} \circ \dots \circ g_n$, for $n=1,2,\dotsc$, where $f_i$
and $g_j$ are holomorphic self-maps of $\mathbb{D}$. Here we obtain a thorough
understanding of such systems $(F_n)$ and $(G_n)$ under the assumptions that
$f_n\to f$ and $g_n\to f$. We determine when the dynamics of $(F_n)$ and
$(G_n)$ mirror that of $(f^n)$, as specified by the Denjoy-Wolff theorem,
thereby providing insight into the stability of the Denjoy-Wolff theorem under
perturbations of the map $f$.
