In the context of a nonequilibrium statistical thermodynamics—based on a nonequilibrium statistical ensemble formalism—a generalized hydrodynamics of fluids under driven flow and shear stress is derived. At the thermodynamic level, the nonequilibrium equations of state are derived, which are coupled to the evolution of the basic variables that describe the hydrodynamic motion in such a system. Generalized diffusion-advection and Maxwell-Cattaneo advection equations are obtained in appropriate limiting situations. This nonlinear higher-order hydrodynamics is applied, in an illustration, to the case of a dilute solution of Brownian particles in nonequilibrium conditions and flowing in a solvent acting as a thermal bath. This is done in the framework of such generalized hydrodynamics but truncated up to a second order.