We have used the free expansion of ultracold neutral plasmas as a time-resolved probe of electron temperature. A combination of experimental measurements of the ion expansion velocity and numerical simulations characterize the crossover from an elastic-collision regime at low initial Gamma_e, which is dominated by adiabatic cooling of the electrons, to the regime of high Gamma_e in which inelastic processes drastically heat the electrons. We identify the time scales and relative contributions of various processes, and experimentally show the importance of radiative decay and disorder-induced electron heating for the first time in ultracold neutral plasmas.