  We study equivariant Gromov-Hausdorff distances for general continuous
actions which are not necessarily isometric as Fukaya introduced. We prove that
if an action is expansive and has pseudo-orbit tracing property then it is
stable under our adapted equivariant Gromov-Hausdorff topology. Finally, using
Lott and Villani's ideas of optimal transport in studying curvature-dimension
conditions, we investigate equivariant Gromov-Hausdorff convergence for actions
of locally compact amenable groups on Wasserstein spaces.
