We report on a growth study of self-catalyzed GaAs nanowires based on time-resolved in situ X-ray structure characterization during molecular-beam-epitaxy in combination with ex situ scanning-electron-microscopy. We reveal the evolution of nanowire radius and polytypism and distinguish radial growth processes responsible for tapering and side-wall growth. We interpret our results using a model for diameter self-stabilization processes during growth of self-catalyzed GaAs nanowires including the shape of the liquid Ga-droplet and its evolution during growth.