Although the stratigraphy of sedimentary basins depends on the balance between the magnitude and grain-size characteristics of the sediment supply (Qs) and the spatial distribution of tectonic subsidence generating accommodation σ(x), Qs is problematical to measure in present-day sediment routing systems and formidably difficult to predict in their ancient counterparts. This challenge was tackled by treating the sediment discharge from the outlet of mountain catchments as the result of incision by a drainage network with a bulk diffusivity based on the length over which the mean annual rainfall is concentrated. The size, relief and slope of palaeo-catchments acting as feeders for sediment routing systems are used to run simulations of sediment discharge and bulk diffusivity for a range of annual precipitation values. A wide range of observable geological phenomena can be used to converge on the most likely solutions for Qs, including depositional volumes in the basin, and bedrock thermochronology and detrital cosmogenic nuclide dating to constrain catchment erosion rates. Modelled sediment discharges can be checked with estimates derived from global regressions. The sediment efflux of mountain catchments serves as a boundary condition for down-system sediment transport and deposition. Variations in the volumetric ratio of sediment supply to available accommodation, Qs/σ(x), determines patterns of transverse versus longitudinal (axial) sediment dispersal. The volumetric ratio may change as a result of variations in climatic parameters, tectonic uplift rate and catchment expansion. An abrupt climate change to higher precipitation values promotes higher Qs/σ(x), but transient landscape response causes a return to values close to the baseline, generating a distinctive down-system extension of a gravel ‘spike’. Catchment expansion has a similar, but more prolonged, effect on gravel progradation. In contrast, a change in tectonic forcing, such as an increase in slip rate on a border fault, causes little change in Qs/σ(x), because increased subsidence compensates for the increased sediment supply. Studies of mid Eocene–Oligocene sediment routing systems in the south-central Pyrenees allow the discrimination of different types of proximal wedge-top sedimentary systems on the basis of the volumetric ratio of Qs to accommodation σ(x): (i) small, steep, local fan systems in tectonically ponded, underfilled basins, supplied by low sediment discharges; (ii) tectonically guided, long-range, axial systems fed by large sediment discharges from widely spaced palaeovalleys; and (iii) large, shallow-sloping transverse megafans burying underlying defunct or active tectonic structures, supplied by high to very high sediment discharges. Understanding the role of variations in Qs helps to explain the syntectonic evolution of proximal foreland basin systems. The Oligocene–Miocene North Alpine Foreland Basin, Switzerland, is qualitatively identified as a high-Qs example, the Miocene–Recent northern Apennines of Italy as a low-Qs example and the Eocene-Oligocene southern Pyrenees of Spain as intermediate in character.