  We introduce a space inhomogeneous generalization of the dynamics on
interlacing arrays considered by Borodin and Ferrari. We show that for a
certain class of initial conditions the point process associated to the
dynamics has determinantal correlation functions and we calculate explicitly,
in the form of a double contour integral, the correlation kernel for one of the
most classical initial conditions, the densely packed. En route to proving this
we obtain some results of independent interest on non-intersecting general
pure-birth chains, that generalize the Charlier process, the discrete analogue
of Dyson's Brownian motion. Finally, these dynamics provide a coupling between
the inhomogeneous versions of the TAZRP and PushTASEP particle systems which
appear as projections on the left and right edges of the array respectively.
