  We study dynamical systems arising as time-dependent compositions of
Pomeau-Manneville-type intermittent maps. We establish central limit theorems
for appropriately scaled and centered Birkhoff-like partial sums, with
estimates on the rate of convergence. For maps chosen from a certain parameter
range, but without additional assumptions on how the maps vary with time, we
obtain a self-normalized CLT provided that the variances of the partial sums
grow sufficiently fast. When the maps are chosen randomly according to a
shift-invariant probability measure, we identify conditions under which the
quenched CLT holds, assuming fiberwise centering. Finally, we show a
multivariate CLT for intermittent quasistatic systems. Our approach is based on
Stein's method of normal approximation.
