We have observed that thin InP nanowires generated by vapor-liquid-solid growth display spontaneous periodic diameter oscillations when large group III supersaturations are used. Diameter variations are associated with a large number of stacking faults and crystallographic phase changes(wurtzite/zinc-blende); also the axial distance between oscillations depends on the indium precursor flow used during the run. We attribute the morphology changes to a substantial deformation of the triple phase line (vapor-liquid-solid) at the catalyst nanoparticle edge originated from multistep nucleation during growth. The deformation alters the mechanical force balance acting on the nanoparticle during growth in such a way that the particle displaces from the nanowire top and wets the nanowire sidewall. Subsequently, as catalytic growth occurs at the sidewall, the associated increase in diameter will eventually push the NP back to its original wire-top position until the onset of a new instability at the triple phase line.