Compositional control in III-V ternary nanowires grown by the vapor-liquid-solid method is essential for bandgap engineering and the design of functional nanowire nano-heterostructures. Herein, we present rather general theoretical considerations and derive explicit forms of the stationary vapor-solid and liquid-solid distributions of vapor-liquid-solid III-V ternary nanowires based on group-III intermix. It is shown that the vapor-solid distribution of such nanowires is kinetically controlled, while the liquid-solid distribution is in equilibrium or nucleation-limited. For a more technologically important vapor-solid distribution connecting nanowire composition with vapor composition, the kinetic suppression of miscibility gaps at a growth temperature is possible, while miscibility gaps (and generally strong non-linearity of the compositional curves) always remain in the equilibrium liquid-solid distribution. We analyze the available experimental data on the compositions of the vapor-liquid-solid Al xGa 1-xAs, In xGa 1-xAs, In xGa 1-xP, and In xGa 1-xN nanowires, which are very well described within the model. Overall, the developed approach circumvents uncertainty in choosing the relevant compositional model (close-to-equilibrium or kinetic), eliminates unknown parameters in the vapor-solid distribution of vapor-liquid-solid nanowires based on group-III intermix, and should be useful for the precise compositional tuning of such nanowires.