A systematic study of exciton dynamics is presented in quantum boxes formed naturally along the axis of a V-shaped quantum wire, by means of time and spatially resolved resonant photoluminescence. The dependence of radiative lifetimes and relaxation mechanisms of excitons is determined versus the size of the boxes. The radiative recombination rate varies linearly with the length of the box, showing that the exciton has a coherence volume equal to the volume of the box. In a low excitation regime, emission from excited states has not been observed, which would be a consequence of relaxation bottleneck, but there is clear evidence that relaxation via emission of LA phonons depends strongly on the energy separation between the different quantum box levels.