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 we experimentally show the importance of radiative decay and disorder-induced electron heating for the first time in ultracold neutral plasmas.