  We study in some generality intertwinings between $h$-transforms of
Karlin-McGregor semigroups associated with one dimensional diffusion processes
and those of their Siegmund duals. We obtain couplings so that the
corresponding processes are interlaced and furthermore give formulae in terms
of block determinants for the transition densities of these coupled processes.
This allows us to build diffusion processes in the space of Gelfand-Tsetlin
patterns so that the evolution of each level is Markovian. We show how known
examples naturally fit into this framework and construct new processes related
to minors of matrix valued diffusions. We also provide explicit formulae for
the transition densities of the particle systems with one-sided collisions at
either edge of such patterns.
