The M–σ relations observed between the masses of central massive objects (CMOs: either a nuclear star cluster or a supermassive black hole) in galaxy nuclei and the stellar velocity dispersion of their host galaxy bulges strongly suggest that the evolution of CMOs and their host galaxies are closely related. Self-regulated feedback from CMOs sweeps the surrounding ambient medium into a shell and when the CMO is at a critical mass the shell is driven from the galaxy, cutting off fuel to the CMO for further growth and locking in the M–σ relations. We investigate the M–σ relations that result from either momentum- or energy-conserving feedback. In the case of momentum-conserving feedback in an isothermal halo, we find the previously derived critical mass is not by itself sufficient to drive the shell to large radius and it takes a CMO with a mass three times the critical value to drive the shell to the escape speed of the halo. In non-isothermal haloes both of these issues are mitigated as the critical mass is sufficient to drive any shell to large radius where it will accelerate and escape the halo. For energy-conserving feedback, we focus on the case that the CMO is a black hole and we find Mvw ∝ σ5, where vw is the black hole wind speed. This relation allows us to infer the wind speeds a sample of now quiescent galaxies would have had during an active phase, and we find good agreement with distributions of observed wind speeds in local active galaxies. We discuss the possibility of a transition from momentum- to energy-driving, the implications of relaxing the assumption of steady CMO winds and the effects these may have on the derived M–σ relations.