Recent experiments reveal spontaneous formation of core-shell heterostructures in different nanowires including InxGa1-xAs, GaxAl1-xAs, and InxGa1-xN. Here, we develop a general growth theory for III-V ternary nanostructures that are fed from a confined reservoir of atoms, and discuss different effects leading to the compositional inhomogeneity and core-shell elemental distributions. In quasi-instantaneous growth of single ternary monolayers AxB1-xC, the material separation can be due to (i) different transport rates of atoms A and B from a mother phase into the solid, (ii) changing the growth environment in favor of C-poor conditions, and (iii) decrease of total supersaturation in the course of monolayer growth. It is shown that scenario (ii) yields abrupt heterointerfaces separating the In-rich or Ga-rich cores from the barrier-type shells in InxGa1-xAs, InxGa1-xP, InxGa1-xN, and GaxAl1-xAs nanowires. The obtained interfacial profiles are primarily determined by the periodically changing V/III ratio of atomic fluxes entering partial monolayer, and are very close for systems with very different pseudobinary interaction parameters. The model fits quite well the interfacial profiles in spontaneous core-shell InxGa1-xAs and InxGa1-xN nanowires, and provides the basis for the future advancements in understanding and controlling these complex effects in different material systems, epitaxy techniques, and nanostructure geometries.