Bed load transport during storm events is both an agent of geomorphic change and a significant natural hazard in mountain regions. Thus, predicting bed load transport is a central challenge in fluvial geomorphology and natural hazard risk assessment. Bed load transport during storm events depends on the width and depth of bed scour, as well as the transport distances of individual sediment grains. We traced individual gravels in two steep mountain streams, the Erlenbach (Switzerland) and Rio Cordon (Italy), using magnetic and radio frequency identification tags, and measured their bed load transport rates using calibrated geophone bed load sensors in the Erlenbach and a bed load trap in the Rio Cordon. Tracer transport distances and bed load volumes exhibited approximate power law scaling with both the peak stream power and the cumulative stream energy of individual hydrologic events. Bed load volumes scaled much more steeply with peak stream power and cumulative stream energy than tracer transport distances did, and bed load volumes scaled as roughly the third power of transport distances. These observations imply that large bed load transport events become large primarily by scouring the bed deeper and wider, and only secondarily by transporting the mobilized sediment farther. Using the sediment continuity equation, we can estimate the mean effective thickness of the actively transported layer, averaged over the entire channel width and the duration of individual flow events. This active layer thickness also followed approximate power law scaling with peak stream power and cumulative stream energy and ranged up to 0.57 m in the Erlenbach, broadly consistent with independent measurements.