This paper reports the results of an experimental and theoretical study into transient granular flows in an inclined, slowly rotating cylinder. The granular materials used were sand, poppy seeds, and glass beads. The following situations have been studied: (i) feeding an initially empty cylinder, (ii) emptying a cylinder which is initially in steady state with equal feed and discharge flow rates, and (iii) the motion of a localized pile of granular material through the cylinder. Axial profiles of bed depth and discharge flow rates have been measured. The mechanistic model used here has previously been used to describe the transient response to large step changes in feed rate, rotation speed and angle of inclination; it has no free parameters. There is good agreement between theory and experiment for sand and poppy seeds, but there are discrepancies for glass beads, which under some conditions flow in a different way from sand and poppy seeds.