Gravity-dominated systems have a negative specific heat. We investigate the negative specific heat of self-gravitating systems enclosed in a spherical container with reflecting walls by means of N-body simulations. To simulate nonequilibrium processes, a particle reflected at a nonadiabatic wall is cooled to mimic energy loss by reflecting walls, while an adiabatic wall is employed for microcanonical ensembles. We show that a negative specific heat occurs not only in the microcanonical ensemble but also in certain nonequilibrium processes with the nonadiabatic wall. With increasing cooling rates, the dependence of temperature T on energy epsilon , i.e., the epsilon-T curve, gradually deviates from the microcanonical ensemble and approaches a certain common curve at a low-energy region. The common curve agrees with an epsilon-T curve for stellar polytropes, especially for the polytrope index of n approximately 5 . We show that the stellar polytrope should be related to the present nonequilibrium process appearing in the self-gravitating system with the nonadiabatic wall. In the nonequilibrium process, a rapid change in velocity at the nonadiabatic wall significantly affects the velocity and density profiles. In particular, the greater the cooling rate, the greater the local velocity gradient at a low-energy region.