We report on the direct observation of the diffusion of carriers in graded InGaN/GaN quantum wells in a nanowire. By probing the local dynamics at the nanoscale, along the wire for different temperatures between 4 and 250 K, we conclude that this diffusion process is thermally activated. In addition, the analysis of the cathodoluminescence lifetime for different temperatures shows that the carrier motion is isotropic and does not follow the indium gradient. Our observations are interpreted in terms of a hopping process between localized states. We find that the random alloy fluctuations prevent any directional drift of excitons along the In gradient and therefore any carrier accumulation. Our results therefore confirm the potential of core shell nanowires for lighting devices. Indeed, the short lifetime of m-plane quantum wells together with their large active area and the homogeneous distribution of carrier along the quantum well will decrease influence of any high carrier density effect on the efficiency of these light-emitting diodes.