We show that the probability of source routing success in dynamic networks, where the link up-down dynamics is governed by a time-varying stochastic process, exhibit critical phase-transition (percolation) phenomena as a function of the end-to-end message latency per unit path length. We evaluate the probability of routing success on dynamic network (1D and 2D) lattices with links going up and down as per an arbitrary binary-valued stationary random process (such as a Markov process), in a source-routing framework. We find percolation thresholds on the time deadline for high-probability of routing success in terms of the first and second order moments of the link state process and we also demonstrate percolation thresholds on the parameters characterizing the link process for a fixed time deadline for a 1D Markov network as an example. This work happens to generalize results reported in two articles from the 80's that appeared in the Physical Review Letters on directed percolation theory. We analyzed the performance of a stateless single-copy opportunistic forwarding algorithm on a 2D probabilistic grid - it does not demonstrate a non-trivial percolation threshold in link-up probability as does a flooding based approach. However, interestingly, when we add a time dimension, i.e., let the network evolve as per a potentially time-correlated link dynamics, the opportunistic store and forward routing algorithm also exhibits a critical threshold behavior. In this 2D grid network case (as in the 1D network case), the normalized messaging latency (ratio of routing latency to path length) exhibits a critical phase transition, where we evaluate the critical latency to path-length ratio as a function of the moments of the link up-down process.