In this article we study the higher-order Moreau's sweeping process introduced in [1], in the case where an exogenous time-varying function u(·) is present in both the linear dynamics and in the unilateral constraints. First we show that the well-posedness results obtained in [1] for the autonomous case, extend to the non-autonomous case when u(·) is analytic, after a suitable state transformation is done. Stability issues are discussed. The complexity of such nonsmooth non-autonomous dynamical systems is illustrated in a particular case named the higher-order bouncing ball, where trajectories with accumulations of jumps are exhibited. Examples from mechanics and circuits illustrate some of the results.