The analysis of realistic numerical simulations of a gel-forming irreversible aggregation process provides information on the role of cluster diffusion in controlling the late stages of the aggregation kinetics. Interestingly, the crossover from chemically controlled to diffusion-controlled aggregation takes place well beyond percolation, after most of the particles have aggregated in the spanning network and only small clusters remain in the sol. The simulation data are scrutinized to gain insight into the origin of this crossover. We show that a single additional time scale (related to the average diffusion time) is sufficient to provide an accurate description of the evolution of the extent of reaction at all times.