Ternary IIIP1-xAsx nanowires and axial heterostructures within such nanowires are perfect building blocks for optoelectronics in an important wavelength range. However, the full control over the morphology and composition of III1-xAsx nanowires as well as the interfacial abruptness of axial In(As)P/InP nanowire hetero-structures is not achieved so far. Here, we present a fully self-consistent model for the growth and composition of gold-catalyzed IIIP1-xAsx nanowires and nanowire heterostructures that explains and predicts interesting phenomena. In particular, we find that the phosphorous concentration in the quaternary Au-III-As-P droplet should be much smaller than that of arsenic under the typical growth conditions. This suppresses the indium crystallization rate when passing from the InAs to InP segment, leading to the droplet swelling. Special care should be taken to keep the droplet on top of the InP segment. We show that the InAs/InP interfacial abruptness is challenging, while the reverse InP/InAs switch should always be atomically sharp. The governing equations connecting thermodynamics and kinetics of ternary nanowires based on IIIP1-xAsx are used to formulate clear routes for the necessary compositional control, with possible extensions to other material systems.