In this paper, we study the asymptotic stability of linear time-varying systems with (sub-) stochastic system matrices. Motivated by applications in distributed dynamic fusion (DDF), we impose some mild regularity conditions on the elements of time-varying system matrices, and provide sufficient conditions under which the asymptotic stability of the underlying LTV system is guaranteed. We partition the sequence of system matrices into non-overlapping slices, and by introducing the notion of unbounded connectivity, we obtain stability conditions in terms of the slice lengths and some network parameters. In particular, we show that the system is asymptotically stable if the unbounded lengths of an infinite subset of slices grow slower than an explicit exponential rate.