The kinetic-energy variation of emitted light clusters has been employed as a clock to explore the time evolution of the temperature for thermalizing composite systems produced in the reactions of 26A, 35A, and 47A MeV Zn64 with Ni58, Mo92, and Au197. For each system investigated, the double-isotope ratio temperature curve exhibits a high maximum apparent temperature, in the range of 10-25 MeV, at high ejectile velocity. These maximum values increase with increasing projectile energy and decrease with increasing target mass. The time at which the maximum in the temperature curve is reached ranges from 80 to 130 fm/c after contact. For each different target, the subsequent cooling curves for all three projectile energies are quite similar. Temperatures comparable with those of limiting temperature systematics are reached 30 to 40 fm/c after the times corresponding to the maxima, at a time when antisymmetrized molecular dynamics transport model calculations predict entry into the final evaporative or fragmentation stage of deexcitation of the hot composite systems. Evidence for the establishment of thermal and chemical equilibrium is discussed.