Sediment influx to channel networks is stochastically driven by rainstorms and other perturbations, which are discrete in time and space and which occur on a landscape with its own spatial variability in topography, colluvium properties, and state of recovery from previous disturbances. The resulting stochastic field of sediment supply interacts with the topology of the channel network and with transport processes to generate spatial and temporal patterns of flux and storage that characterize the sedimentation regime of a drainage basin. The regime varies systematically with basin area. We describe how the stochastic sediment supply is generated by climatic, topographic, geotechnical, and biotic controls that vary between regions. The general principle is illustrated through application to a landscape where sediment is supplied by mass wasting, and the forcing variables are deterministic thickening of colluvium, random sequences of root‐destroying wildfires, and random sequences of rainstorms that trigger failure in a population of landslide source areas with spatial variance in topography and colluvium strength. Landslides stop in channels or convert to scouring debris flows, depending on the nature of the low‐order channel network. Sediment accumulates within these channels for centuries before being transferred downstream by debris flows. Time series of sediment supply, transport, and storage vary with basin scale for any combination of climatic, topographic, and geotechnical controls. In a companion paper [Benda and Dunne, this issue] we use simulations of timing, volumes, and locations of mass wasting to study the interaction between a stochastically forced sediment supply and systematic changes of storage and flux through channel networks.