  In this paper we advance the entropy theory of discrete nonautonomous
dynamical systems that was initiated by Kolyada and Snoha in 1996. The first
part of the paper is devoted to the measure-theoretic entropy theory of general
topological systems. We derive several conditions guaranteeing that an initial
probability measure, when pushed forward by the system, produces an invariant
measure sequence whose entropy captures the dynamics on arbitrarily fine
scales. In the second part of the paper, we apply the general theory to the
nonstationary subshifts of finite type, introduced by Fisher and Arnoux. In
particular, we give sufficient conditions for the variational principle,
relating the topological and measure-theoretic entropy, to hold.
