We investigate the axial and radial growth of GaN nanowires upon a variation of the Ga flux during molecular beam epitaxial growth. An increase in the Ga flux promotes radial growth without affecting the axial growth rate. In contrast, a decrease in the Ga flux reduces the axial growth rate without any change in the radius. These results are explained by a kinetic growth model that accounts for both the diffusion of Ga adatoms along the side facets toward the nanowire tip and the finite amount of active N available for the growth. The model explains the formation of a new equilibrium nanowire radius after increasing the Ga flux and provides an explanation for two well-known but so far not understood experimental facts: the necessity of effectively N-rich conditions for the spontaneous growth of GaN nanowires and the increase in nanowire radius with increasing III/V flux ratio.