The general problem of adiabatic relaxation, or thermal equilibration between a sample and a reservoir initially at two different temperatures, is presented. By thermostatting both the sample and reservoir, a nonequilibrium steady state can be set up to measure this relaxation rate. An accurate treatment of the coupling between sample and reservoir in the steady-state case leads to substantially improved agreement with vibrational relaxation rates obtained from adiabatic equilibration. We show that for large signal to noise ratio, the nonequilibrium simulation can be considerably more accurate than the direct equilibration measurement. We demonstrate two other examples of transport phenomena obtained from thermostatted sample-reservoir driving, namely, thermal conduction and shear flow in a fluid.