River currents, wind, and waves drive bed-load transport, in which sediment particles collide with each other and Earth's surface. A generic consequence is impact attrition and rounding of particles as a result of chipping, often referred to in geological literature as abrasion. Recent studies have shown that the rounding of river pebbles can be modeled as diffusion of surface curvature, indicating that geometric aspects of impact attrition are insensitive to details of collisions and material properties. We present data from fluvial, aeolian, and coastal environments and laboratory experiments that suggest a common relation between circularity and mass attrition for particles transported as bed load. Theory and simulations demonstrate that universal characteristics of shape evolution arise because of three constraints: (i) Initial particles are mildly elongated fragments, (ii) particles collide with similarly-sized particles or the bed, and (iii) collision energy is small enough that chipping dominates over fragmentation but large enough that sliding friction is negligible. We show that bed-load transport selects these constraints, providing the foundation to estimate a particle's attrition rate from its shape alone in most sedimentary environments. These findings may be used to determine the contribution of attrition to downstream fining in rivers and deserts and to infer transport conditions using only images of sediment grains.