  We study Markov processes where the "time" parameter is replaced by paths in
a directed graph from an initial vertex to a terminal one. Along each directed
path the process is Markov and has the same distribution as the one along any
other directed path. If two directed paths do not interact, in a suitable
sense, then the distributions of the processes on the two paths are
conditionally independent, given their values at the common endpoint of the two
paths. Conditions on graphs that support such processes (e.g., hexagonal
lattice) are established. Next we analyze a particularly suitable family of
Markov processes, called harnesses, which includes Brownian motion and other
L\'{e}vy processes, on such time-like graphs. Finally we investigate continuum
limits of harnesses on a sequence of time-like graphs that admits a limit in a
suitable sense.
