While many rheological studies are performed at a fixed concentration, most granular flows are constrained, not by concentration, but by an applied stress. The stress constraint sets the average concentration, but the material is free to vary that concentration slightly to match the applied stress with that generated internally. This study examines stress-controlled systems in light of recent findings that the elastic properties of the particles appear as constitutive parameters even in flowing situations. Stress-controlled flows are shown to behave very differently from flows at fixed concentration. In particular, if the stress is fixed and the shear rate is slowly increased, the flow exhibits the expected progression from elastic–quasi-static to elastic–inertial to inertial flow – a sequence opposite to that followed in fixed-concentration flows. Thus system-scale constraints can have a profound effect on granular rheology.