Bed load sediment particles supplied to channels by hillslopes are reduced in size by abrasion during downstream transport. The branching structure of the channel network creates a distribution of downstream travel distances to a given reach of river and thus may strongly influence the grain size distribution of the long-term bed load flux through that reach. Here we investigate this hypothesis, using mass conservation and the Sternberg exponential decay equation for particle abrasion, to predict bed material variability at multiple scales for both natural and artificial drainage networks. We assume that over a sufficiently long timescale, no net deposition occurs and that grains less than 2 mm are swept away in suspension. We find that abrasion during fluvial transport has a surprisingly small effect on the bed load sediment grain size distribution, for the simple case of spatially uniform supply of poorly sorted hillslope sediments. This occurs because at any point in the channel network, local resupply offsets the size reduction of material transported from upstream. Thus river bed material may essentially mirror the coarse component of the size distribution of hillslope sediment supply. Furthermore, there is a predictable distance downstream at which the bed load grain size distribution reaches a steady state. In the absence of net deposition due to selective transport, large-scale variability in bed material, such as downstream fining, must then be due primarily to spatial gradients in hillslope sediment production and transport characteristics. A second key finding is that average bed load flux will tend to stabilize at a constant value, independent of upstream drainage area, once the rate of silt production by bed load abrasion per unit travel distance is equal to the rate of coarse sediment supply per unit channel length (q). Bed load flux equilibrates over a distance that scales with the inverse of the fining coefficient in the abrasion rate law (α) and can be approximated simply as q/3α. Thus the efficiency of particle abrasion sets a fundamental length scale, shorter for weaker rocks and longer for harder rocks, which controls the expression in the river bed of variability in sediment supply. We explore the role of the abrasion length scale in modulating the influence of sediment supply variability in a number of channel network contexts, including individual tributary junctions, a sequence of tributary inputs along a main stem channel, and variable basin shapes and network architecture as expressed by the width function. These findings highlight the need for both data and theory that can be used to predict the grain size distributions supplied to channels by hillslopes.