We study the dynamics of the cooling of a gas of caesium atoms in an optical lattice, both experimentally and with 1D full-quantum Monte Carlo simulations. We find that, contrary to the standard interpretation of the Sisyphus model, the cooling process does not work by a continuous decrease of the average kinetic energy of the atoms in the lattice. Instead, we show that the momentum of the atoms follows a bimodal distribution, the atoms being gradually transferred from a hot to a cold mode. We suggest that the cooling mechanism should be depicted in terms of a rate model, describing the transfer between the two modes along with the processes occurring within each mode.