The Jarzynski equality and the Crooks fluctuation theorem enable the calculation of the change in a system's free energy from nonequilibrium path integrals. These relations consider processes where the system is driven out of equilibrium by a mechanical external agent while remaining in contact with a thermal reservoir at a fixed temperature. We generalize these relations to describe processes driven by any type of external agent, be it thermal or mechanical. Attention is given to the case of a system, initially in equilibrium, that is driven through a temperature change by a heat reservoir. The results are cast in a form applicable to experiments.