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We consider asymptotic properties of families of multifunctions generated by cocycles, families of measures as well as Markov operators associated with such a mapping. The results we apply to random dynamical systems show connection of obtained attractors for families of multifunctions with global set attractors and also connection of supports of attracting measures for Markov families with global point attractors. The use of topological limits instead of standard Hausdorff distance lets us to get attractors under some quite general assumptions only on a cocycle mapping but without any assumptions on so-called parameter space and without standard assumption on existence of a compact absorbing set. [ABSTRACT FROM AUTHOR]