This contribution reports on the mass transport kinetics of osmotically imbalanced water-in-oil-in-water (W1/O/W2) emulsions. Although frequently studied, the control of mass transport in W1/O/W2 emulsions is still challenging. We describe a microfluidics-based method to systematically investigate the impact of various parameters, such as osmotic pressure gradient, oil phase viscosity, and temperature, on the mass transport. Combined with optical microscopy analyses, we are able to identify and decouple the various mechanisms, which control the dynamic droplet size of osmotically imbalanced W1/O/W2 emulsions. So, swelling kinetics curves with a very high accuracy are generated, giving a basis for quantifying the kinetic aspects of transport. Two sequential swelling stages, i.e., a lag stage and an osmotically dominated stage, with different mass transport mechanisms are identified. The determination and interpretation of the different stages are the prerequisite to control and trigger the swelling process. We show evidence that both mass transport mechanisms can be decoupled from each other. Rapid osmotically driven mass transport only takes place in a second stage induced by structural changes of the oil phase in a lag stage, which allow an osmotic exchange between both water phases. Such structural changes are strongly facilitated by spontaneous water-in-oil emulsification. The duration of the lag stage is pressure-independent but significantly influenced by the oil phase viscosity and temperature.