  We consider consistent dynamics for non-intersecting birth and death chains,
originating from dualities of stochastic coalescing flows and one dimensional
orthogonal polynomials. As corollaries, we obtain unified and simple
probabilistic proofs of certain key intertwining relations between multivariate
Markov chains on the levels of some branching graphs. Special cases include the
dynamics on the Gelfand-Tsetlin graph considered by Borodin and Olshanski and
the ones on the BC-type graph recently studied by Cuenca. Moreover, we
introduce a general inhomogeneous random growth process with a wall that
includes as special cases the ones considered by Borodin and Kuan and Cerenzia,
that are related to the representation theory of classical groups and also the
Jacobi growth process more recently studied by Cerenzia and Kuan. Its most
important feature is that, this process retains the determinantal structure of
the ones studied previously and for the fully packed initial condition we are
able to calculate its correlation kernel explicitly in terms of a contour
integral involving orthogonal polynomials. At a certain scaling limit, at a
finite distance from the wall, one obtains for a single level discrete
determinantal ensembles associated to continuous orthogonal polynomials, that
were recently introduced by Borodin and Olshanski, and that depend on the
inhomogeneities.
