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Understanding the macroscopically observable behavior of large complex systems based on the laws governing their microscopic constituents is an intriguing fundamental problem. Traditional statistical mechanics introduces "ensembles" (e.g., microcanonical, canonical, ...) to describe many-body systems at equilibrium. The success of this framework relies on the empirical fact that many microscopically distinct systems or states exhibit the same macroscopic properties. We present an approach to adapt similar ideas to the dynamics of many-body quantum systems. Specifically, we derive analytical predictions for the time dependence of observable expectation values in two nonequilibrium setups: relaxation under the influence of perturbations and so-called echo protocols involving an effective time reversal to probe for signatures of classical chaos. We also verify those predictions by comparison with experiments and numerical simulations.