In this paper, we address the stability of non-autonomous difference equations by providing a suitable representation of the solution at time $t$ in terms of the initial condition and time-dependent matrix coefficients. This enables us to characterize the asymptotic behavior of solutions in terms of that of such coefficients. As a consequence, we obtain necessary and sufficient stability criteria for non-autonomous linear difference equations. In the case of difference equations with arbitrary switching, we obtain a generalization of the well-known criterion for autonomous systems due to Hale and Silkowski, which, as the latter, is delay-independent. These results are applied to transport and wave propagation on networks. In particular, we show that the wave equation on a network with arbitrarily switching damping at external vertices is exponentially stable if and only if the network is a tree and the damping is bounded away from zero at all external vertices but one.