In this paper, we consider the problem of oil molecular transport between nonequilibrium microemulsion oil droplets in water. In particular, we have investigated the kinetics of solubilization of big oil drops by smaller microemulsion droplets transforming a bimodal size distribution into an equilibrium microemulsion with a unimodal size distribution. The process involves the diffusion of oil monomers across the aqueous solvent. Solubilization experiments are presented on a well-characterized nonionic microemulsion system, where interfacial area and dispersed volume are conserved and where the excess oil chemical potential is dominated by interfacial curvature energy. An analysis of the experiments shows that the solubilization kinetics depend significantly on the concentrations and sizes of big and small droplets. We formulate a quantitative model for the solubilization kinetics where the effects of size and concentrations are treated within the framework of a cell model. A quantitative agreement between model and experiment is obtained, and the analysis also shows that the majority of oil monomers are captured by small droplets already in the vicinity of the big drop surface when the concentration of small droplets is high.