Set-valued dynamical systems in discrete time as iterations of a multi-function (relation), or dispersive ows in continuous time, have attracted the attention of researchers during the last decades. Most of these works were dedicated to the topology of limit sets, including geometry of the attractors. Much less attention was paid to the dynamics itself on these limit sets. Chain recurrence in dispersive ows has been considered by I. U. Bronshteyn and A. Y. Kopanski (1984, 1985). D. N. Cheban (2010) has studied the limit sets in dissipative such ows. E. Akin (e.g., 1993, 2017) studied various aspects of set valued dynamics and their topological properties, including Shadowing property. B. E. Raines and T. Tennant (2015) have studied set-valued dynamical systems and their inverse limits, having the Speci cation property. The dynamics in its own rights of some (weakly) contracting with respect to the Pompeiu-Hausdor metric relations in complete metric spaces has been considered by the authors (2003, 2004). More precisely, we stated the "asymptotic phase" property, topological transitivity on the attractor as well as Shadowing property near the attractor. In our report we strengthen the transitivity and Shadowing properties up to topological mixing and Speci cation property, respectively. Following M. Hata and I. A. Rus, we relax the contractivity condition for set-valued mappings to a level of weak contractivity, using the notion of comparison function.