By means of kinetic Monte Carlo simulations of the self-organized growth of quantum dots in strained semiconductor systems we resolve the seemingly contradictory features of kinetic versus thermodynamic behavior, e.g., with respect to the temperature dependence of the average dot size and their dispersion. We show that the size distribution immediately after deposition is kinetically controlled, with smaller islands for lower temperatures and larger islands for higher temperatures. For longer simulation times the kinetics leads to equilibration, and a crossover effect between the size distributions occurs, which is in good agreement with the predictions of thermodynamics.