  A method to approximately close the dynamic cavity equations for synchronous
reversible dynamics on a locally tree-like topology is presented. The method
builds on $(a)$ a graph expansion to eliminate loops from the normalizations of
each step in the dynamics, and $(b)$ an assumption that a set of auxilary
probability distributions on histories of pairs of spins mainly have
dependencies that are local in time. The closure is then effectuated by
projecting these probability distributions on $n$-step Markov processes. The
method is shown in detail on the level of ordinary Markov processes ($n=1$),
and outlined for higher-order approximations ($n>1$). Numerical validations of
the technique are provided for the reconstruction of the transient and
equilibrium dynamics of the kinetic Ising model on a random graph with
arbitrary connectivity symmetry.
