Let (X,f) be a dynamical system, where X is a nondegenerate continuum and f is a map. For any positive integer n, we consider the hyperspace Fn(X) with the Vietoris topology. For n>1 and K∈Fn(X) the subset Fn(K,X) of Fn(X) is defined as the collection of elements of Fn(X) containing K. We consider the quotient hyperspace FnK(X)=Fn(X)⧸Fn(K,X), which is obtained from Fn(X) by shrinking Fn(K,X) to one point set. Furthermore, we consider the induced maps Fn(f):Fn(X)→Fn(X) and FnK(f):FnK(X)→FnK(X). In this paper, we introduce the dynamical system (FnK(X),FnK(f)) and we study relationships between the conditions f∈M, Fn(f)∈M and FnK(f)∈M, where M is one of the following classes of maps: transitive, mixing, weakly mixing, totally transitive, exact, exact in the sense of Akin-Auslander-Nagar, strongly transitive in the sense of Akin-Auslander-Nagar, exact transitive, fully exact, strongly exact transitive, strongly product transitive, orbit-transitive, Devaney chaotic, irreducible, TT++, strongly transitive and very strongly transitive.