Water flowing over a loose granular bed organizes into a braided river, a network of ephemeral and interacting channels. The temporal and spatial evolution of this network of braided channels is not yet quantitatively understood. In ∼1 m-scale experiments, we found that individual channels exhibit a self-similar geometry and near-threshold transport conditions. Measurements of the rate of growth of topographic correlation length scales, the time scale of system-slope establishment, and the random spatial decorrelation of channel locations indicate together that the evolution of the braided river system may be diffusive in nature. This diffusion is due to the separation of scales between channel formation and network evolution, and the random motion of interacting channels when viewed at a coarse-grained scale.