One important class of nonequilibrium systems involves quenching, so that the system evolves towards a new equilibrium state -- for instance the phase separation of binary fluids. Another class involves driving, so that the system is maintained out of equilibrium indefinitely -- for instance by shear. Of special interest are systems where both features are present simultaneously. An important question in such cases is: does the system reach a nonequilibrium steady state as a result of the drive, or does it continue to evolve as a result of the quench? For sheared binary fluids, simple scaling arguments and also analytic approximations cannot answer this question. I will present extensive lattice Boltzmann simulations in both two and three dimensions that do seem to resolve the issue. Related simulations on other nonequilibrium systems, such as active liquid crystals, may also be briefly discussed.