The dynamics of the diffusion-limited model of cluster-cluster aggregation is investigated in two and three dimensions by studying the temporal evolution of the cluster-size distribution ${n}_{s}$(t), which is the number of clusters of size s at time t. In a recent study it was shown that the results of the two-dimensional simulations for mass-independent diffusivity can be well represented by a dynamic-scaling function of the form ${n}_{s}$(t)\ensuremath{\sim}${s}^{\mathrm{\ensuremath{-}}2}$f(s/${t}^{z}$), where f(x) is a scaling function with a power-law behavior for small x, namely f(x)\ensuremath{\sim}${x}^{\ensuremath{\delta}}$ for x\ensuremath{\ll}1 and f(x)\ensuremath{\ll}1 for x\ensuremath{\gg}1. In this paper we extend the calculations of the cluster-size distribution to three dimensions and to the case of the cluster diffusivity depending on the size of the clusters. The diffusion constant of a cluster of size s is assumed to be proportional to ${s}^{\ensuremath{\gamma}}$. The overall behavior of ${n}_{s}$(t) and the exponents \ensuremath{\delta} and z have been determined for a set of values of \ensuremath{\gamma}. We find that the results are consistent with the scaling theory, and the exponents in ${n}_{s}$(t) depend continuously on \ensuremath{\gamma}. Moreover, there is a critical value of \ensuremath{\gamma} [${\ensuremath{\gamma}}_{c}$(d=2)\ensuremath{\simeq}-(1/4), ${\ensuremath{\gamma}}_{c}$(d=3)\ensuremath{\simeq}-1/2] at which the shape of the cluster-size distribution crosses over from a monotonically decreasing function to a bell-shaped curve which can be described by the above scaling form for ${n}_{s}$(t), but with a scaling function f\ifmmode \tilde{}\else \~{}\fi{}(x) different from f(x).