  We adapt Stein's method to obtain Berry--Esseen type error bounds in the
multivariate central limit theorem for non-stationary processes generated by
time-dependent compositions of uniformly expanding dynamical systems. In a
particular case of random dynamical systems with a strongly mixing base
transformation, we derive an error estimate of order $O(N^{-1/2})$ in the
quenched multivariate CLT, provided that the covariance matrix "grows linearly"
with the number of summands $N$. The error in the normal approximation is
estimated for the class of all convex sets.
