We derive a theoretical model for the nonequilibrium stress in a flowing colloidal suspension by tracking the motion of a single embedded probe. While Stokes-Einstein relations connect passive, observable diffusion of a colloidal particle to properties of the suspending medium, they are limited to linear response. Actively forcing a probe through a suspension produces nonequilibrium stress that at steady state can be related directly to observable probe motion utilizing an equation of motion rather than an equation of state, giving a nonequilibrium Stokes-Einstein relation [J. Rheol., 2012, 56, 1175-1208]. Here that freely-draining theory is expanded to account for hydrodynamic interactions. To do so, we construct an effective hydrodynamic resistance tensor, through which the particle flux is projected to give the advective and diffusive components of a Cauchy momentum balance. The resultant phenomenological relation between suspension stress, viscosity and diffusivity is a generalized nonequilibrium Stokes-Einstein relation. The phenomenological model is compared with the statistical mechanics theory for dilute suspensions as well as dynamic simulation at finite concentration which show good agreement, indicating that the suspension stress, viscosity, and force-induced diffusion in a flowing colloidal dispersion can be obtained simply by tracking the motion of a single Brownian probe. (C) 2018 Elsevier Inc. All rights reserved.