A continuous exchange of particles between an erodible substrate and the granular flow above it occurs during almost all geophysical events involving granular material, such as snow avalanches, debris flows and pyroclastic flows. The balance between eroded and deposited material can drastically influence the runout distance and duration of the flow. In certain conditions, a perfect balance between erosion and deposition may occur, leading to the steady propagation of material, in which the flow maintains its shape and velocity throughout. It is shown experimentally how the erosion-deposition process in dense flows of sand (160-200 μm) on an erodible bed of the same material produces steadily propagating avalanches that deposit subtle levees at their lateral extent. Moreover, it is shown in this paper, by using two colours of the same sand, that although the avalanche is propagating at constant velocity and maintaining a constant shape, the grains that are initially released are deposited along the flow path and that the avalanche will eventually be composed entirely of particles that are eroded from the bed. Different steady travelling wave regimes are obtained depending on the slope angle, thickness of the erodible layer and the amount of material released. Outside of the range of parameters where steady travelling waves form, the avalanches loose mass and decay if the initial amount of material released is too small, or, if the initial release is too large, they re-adjust to a steadily propagating regime by shedding material and breaking into smaller avalanches at its rear side. Numerical simulations are performed using a shallow-water-like avalanche model together with a friction law that captures the erosion-deposition process in flowing to static regimes and a transport equation for the interface between layers of the two colours. The characteristic behaviours observed in the experiments are qualitatively reproduced. Specifically, the complex processes such as the exchange of particles leading to a change in colour of the avalanche and the formation of lateral levees are captured by the model. Finally a comparison is made with deposits in lunar craters, which are interpreted as closely analogous to the deposits formed in our laboratory experiments.