Mechanical signals are believed to play a major role in organizing the collective motility of epithelial cell clusters on a substrate. A number of experimental observations in these systems await a comprehensive explanation: the interior is tensile even for clusters that expand by proliferation; the tractions on the substrate are often confined to the cluster edges; mechanical waves can propagate within the cluster; cells can spontaneously fill an annulus by proliferation and initiate unidirectional rotation around it; cell clusters can durotax much more efficiently than individual cells. We formulate a mechanical model to examine these effects. We include cell motility cycles comprised of active contraction and protrusion, and use a molecular clutch picture allowing “stalling” --- inhibition of cell contraction by external forces. By attaching cells to the substrate and to each other, and taking into account contact inhibition of locomotion, we obtain a simple picture underlying many of these findings.