Exploring a recently developed mesoscale continuum theory of dislocation dynamics, we derive three predictions about plasticity and grain boundary formation in crystals. (1) There is a residual stress jump across grain boundaries and plasticity-induced cell walls as they form, which self-consistently acts to attract neighboring dislocations; residual stress in this theory appears as a remnant of the driving force behind wall formation under both polygonization and plastic deformation. We derive the predicted asymptotic late-time dynamics of the grain-boundary formation process. (2) During grain boundary formation at high temperatures, there is a predicted cusp in the elastic energy density. (3) In early stages of plasticity, when only one type of dislocation is active (single-slip), cell walls do not form in the theory; instead we predict the formation of a hitherto unrecognized jump singularity in the dislocation density.