We extend a quantum kinetic approach to the description of hadronic showers in space, time, and momentum space to deep-inelastic $\mathrm{ep}$ collisions, with particular reference to experiments at DESY HERA. We follow the history of hard scattering events back to the initial hadronic state and forward to the formation of color-singlet prehadronic clusters and their decays into hadrons. The time evolution of the spacelike initial-state shower and the timelike secondary partons are treated similarly, and cluster formation is treated using a spatial criterion motivated by confinement and a nonperturbative model for hadronization. We calculate the time evolution of particle distributions in rapidity, transverse, and longitudinal space. We also compare the transverse hadronic energy flow and the distribution of observed hadronic masses with experimental data from HERA, finding encouraging results, and discuss the background to large-rapidity-gap events. The techniques developed in this paper may be applied in the future to more complicated processes such as $\mathrm{eA}$, $\mathrm{pp}$, $\mathrm{pA}$, and $\mathrm{AA}$ collisions.
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