  Recently, there has been an increasing interest on nonautonomous composition
of perturbed hyperbolic systems: composing perturbations of a given hyperbolic
map $F$ results in statistical behaviour close to that of $F$. We show this
fact in the case of piecewise regular expanding maps. In particular, we impose
conditions on perturbations of this class of maps that include situations
slightly more general than what has been considered so far, and prove that
these are stochastically stable in the usual sense. We then prove that the
evolution of a given distribution of mass under composition of time dependent
perturbations (arbitrarily - rather than randomly - chosen at each step) close
to a given map $F$ remains close to the invariant mass distribution of $F$.
Moreover, for almost every point, Birkhoff averages along trajectories do not
fluctuate wildly. This result complements recent results on memory loss for
nonautonomous dynamical systems.
