Sudden topological rearrangement of neighboring bubbles in a foam occur during coarsening, and can also be induced by applied forces. Diffusing-wave spectroscopy measurements are presented of such dynamics before, during, and after an imposed shear strain. The rate of rearrangements is proportional to the strain rate, and the shape of the correlation functions shows that they are spatially and temporally uncorrelated. Macroscopic deformation is thus accomplished by a nonlinear microscopic process reminiscent of dynamics in the propagation of earthquake faults or the flow of granular media.