Peculiar velocities of galaxies reflect continued action of gravity working throughout the whole cosmic expansion history. This indicates that velocity-based probes and observational statistics are, in principle, very well posed in principle to be sensitive to even very moderate modifications to the standard laws of GR. Whether it be a theory of modified gravity (MOG) or a model of an interacting Dark Energy, the cosmic galaxy velocity field should be still sensitive to produced deviation from standard growth of structures scenario. I am going to present a class of velocity-based observational statistics that can be used to test GR on intermediate cosmological scales. I will also discuss weak and strong points of various probes and give some prospects for the future.