Kinetic modeling of the formation of axial heterostructures in III-V nanowires grown by the Au-catalyzed vapor-liquid-solid method is presented. We link the mass balance in the droplet to the crystallization rates of different III-V pairs at the liquid-solid interface. This allows us to describe how the chemical composition changes across a nanowire heterostructure and study the influence of the growth parameters on the interfacial abruptness. It is shown that, at high enough supersaturation in liquid, there is no segregation of different binaries in solid even for materials systems with strong interactions between III-V pairs, such as InGaAs. This leads to the suppression of the miscibility gaps by kinetic factors. The influence of the Au concentration on the interfacial abruptness is found to be complicated. Increasing the Au concentration widens the heterointerface at low Au content and narrows it at high Au content in a catalyst droplet. The model fits quite well the data on the compositional profiles across axial nanowire heterostructures based on both group III and group V interchange. Very sharp heterointerfaces in double of InAs/InP/InAs nanowire heterostructures is explained by a reduced reservoir effect due to low solubility of group V elements in the droplet.