The apicobasal polarity of epithelia depends on the integrated activity of apical and basolateral proteins, and is essential for tissue integrity and body homeostasis. Yet these tissues are frequently on the move as they are sculpted by active morphogenetic cell rearrangements. How does cell polarity survive these stresses? We analyse this question in the renal tubules of Drosophila, a tissue that undergoes dramatic morphogenetic change as it develops. Here we show that, whereas the Bazooka and Scribble protein groups are required for the establishment of tubule cell polarity, the key apical determinant, Crumbs, is required for cell polarity in the tubules only from the time when morphogenetic movements start. Strikingly, if these movements are stalled, polarity persists in the absence of Crumbs. Similar rescue of the ectodermal phenotype of the crumbs mutant when germ-band extension is reduced suggests that Crumbs has a specific, conserved function in stabilising cell polarity during tissue remodelling rather than in its initial stabilisation. We also identify a requirement for the exocyst component Exo84 during tissue morphogenesis, which suggests that Crumbs-dependent stability of epithelial polarity is correlated with a requirement for membrane recycling and targeted vesicle delivery.