  We consider families of fast-slow skew product maps of the form
\begin{align*} x_{n+1} = x_n+\epsilon a(x_n,y_n,\epsilon), \quad y_{n+1} =
T_\epsilon y_n, \end{align*} where $T_\epsilon$ is a family of nonuniformly
expanding maps, and prove averaging and rates of averaging for the slow
variables $x$ as $\epsilon\to0$. Similar results are obtained also for
continuous time systems \begin{align*} \dot x = \epsilon a(x,y,\epsilon), \quad
\dot y = g_\epsilon(y). \end{align*} Our results include cases where the family
of fast dynamical systems consists of intermittent maps, unimodal maps (along
the Collet-Eckmann parameters) and Viana maps.
