<p>It is well known that the topological entropy of the induced hyperspace map is positive, provided so is the topological entropy of the original continuous single-valued map. In this paper we ask whether this is also true for the sequences of multivalued maps. In particular, we consider various definitions (some of them for the first time) of topological entropy for multivalued nonautonomous dynamical systems in compact Hausdorff spaces and investigate their mutual relations. We explain that some of them can deal with arbitrary multivalued maps, i.e. without regularity restrictions. For upper semicontinuous multivalued maps, we study the parametric topological entropy of the induced possibly discontinuous dynamical systems in hyperspaces. We show, among other things, that a comparison of different types of entropy notions allows us to make the entropy estimates without direct calculations. Several illustrative examples are supplied.</p>