- We study the evolution of the energy distribution for a stadium with moving walls. We consider a one period driving cycle, which is characterized by an amplitude A and a wall velocity V. This evolving energy distribution has both ``parametric'' and ``stochastic'' components. The latter are important for the theory of quantum irreversibility and dissipation in driven mesoscopic devices. For an extremely slow wall velocity V, the spreading mechanism is dominated by transitions between neighboring levels, while for larger (nonadiabatic) velocities, the spreading mechanism has both perturbative and nonperturbative features. We present a numerical study which is aimed at identifying the latter features. A procedure is developed for the determination of the various V regimes. The possible implications of linear response theory are discussed.