Nanoscale membranes have emerged as a new class of vertical nanostructures that enable the integration of horizontal networks of III-V nanowires on a chip. To generalize this method to the whole family of III-Vs, progress in the understanding of the membrane formation by selective area epitaxy in oxide slits is needed, in particular for different slit orientations. Here, it is demonstrated that the shape is primarily driven by the growth kinetics rather than determined by surface energy minimization as commonly occurs for faceted nanostructures. To this end, a phase-field model simulating the shape evolution during growth is devised, in agreement with the experimental findings for any slit orientations, even when the vertical membranes turn into multifaceted fins. This makes it possible to reverseengineer the facet-dependent incorporation times, which were so far unknown, even for common low-index facets. The compelling reproduction of the experimental morphologies demonstrates the reliability of the growth model and offers a general method to determine microscopic kinetic parameters governing out-of-equilibrium three-dimensional growth.