Technical Abstract: The local-uniform-flow (LUF) hypothesis is a simplification of the governing equations describing river morphodynamics, which is needed to determine the evolution of the bed profile and bed-material composition in the case of large time and space scales. This paper presents a rigorous analysis of the full one-dimensional river hydrodynamic and morphodynamic mathematical model and its LUF approximation. The analysis establishes two criteria to assess the validity of the LUF hypothesis: (1) a criterion for rivers in equilibrium and (2) a criterion for evolving rivers (i.e., in non-equilibrium). The first criterion is based on the concept of the morphological box. Variations of the river bed longer than the box length are adequately reproduced by the LUF hypothesis, whereas only spatially averaged values are resolved within the box. The second criterion is based on the concept of an evolution window. Temporal variations represented by sinusoidal waves with wave periods larger than the evolution window can be adequately reproduced by the LUF hypothesis, whereas variations with shorter periods are averaged within this window. To limit the error introduced by the LUF hypothesis the minimum size of the morphological box and the evolution window increases when the Froude number decreases. Further, the minimum size of the evolution window increases for decreasing sediment concentration and increasing mixing layer thickness (i.e., for larger bed forms). The LUF hypothesis is therefore best applied to small mountain rivers for which both the minimum size of the morphological box and the evolution window is relatively small, so that spatial and temporal variations can be resolved in more detail. Applications using the LUF hypothesis for large watersheds (including the lowland portion of the fluvial network) are possible, but are limited to simulations over larger spatial and temporal intervals.