Burago et al. (Graduate studies in mathematics, AMS, 2001) considered quotient metric spaces consisting of orbits with respect to some isometry groups. We extend their approach over some semigroups of transformations. We are concerned with quotient semi-metrics in spaces of generalized orbits and give conditions sufficient for these semi-metrics to be metrics. We apply our approach to hyperspaces of compact convex subsets of Euclidean n-space and to that of convex bodies, endowed, respectively, with the Hausdorff metric and with the symmetric difference metric.