Alluvial river networks are key components of sedimentary systems. They transport sediment, supplied from eroding source regions, to downstream depositional sinks. Their slopes and the rates at which they transport sediment are controlled by their water and sediment supplies, which are sensitive to environmental conditions: temperature and precipitation rates influence erosion rates, and hence sediment supply; precipitation rates are also a first order control on water supply. Changes in sediment and water supply therefore drive changes in channel slope, which may be preserved in fluvial terrace sequences, and sediment-transport rates, which may be preserved in downstream stratigraphic records. Consequently, if we can understand this behaviour, we may be able to use geomorphic and stratigraphic archives to reconstruct past environmental change and its influence on landscapes. Here, we apply a physically based model describing the coupled evolution of sediment transport and slope along alluvial rivers. We show that aggradation, incision, and variation in sediment-transport rates can be damped and lag significantly behind environmental change, depending on the forcing timescale, and on network geometry and hydrology. When water supply is varied, variation in sediment output can in some cases be amplified and appear to lead the imposed forcing. These effects should be taken into account when interpreting fluvial terrace and sedimentary records. Contrasting patterns of aggradation and incision compared with sediment-transport rates implies that integrating information from geomorphic and stratigraphic archives will provide deeper paleo-climatic insights and a powerful test of our understanding of sedimentary systems.