  We consider consistent diffusion dynamics, leaving the celebrated
Hua-Pickrell measures, depending on a complex parameter $s$, invariant. These,
give rise to Feller-Markov processes on the infinite dimensional boundary
$\Omega$ of the "graph of spectra", the continuum analogue of the
Gelfand-Tsetlin graph, via the method of intertwiners of Borodin and Olshanski.
In the particular case of $s=0$, this stochastic process is closely related to
the $\mathsf{Sine_2}$ point process on $\mathbb{R}$ that describes the spectrum
in the bulk of large random matrices. Equivalently, these coherent dynamics are
associated to interlacing diffusions in Gelfand-Tsetlin patterns having certain
Gibbs invariant measures. Moreover, under an application of the Cayley
transform when $s=0$ we obtain processes on the circle leaving invariant the
multilevel Circular Unitary Ensemble. We finally prove that the Feller
processes on $\Omega$ corresponding to Dyson's Brownian motion and its
stationary analogue are given by explicit and very simple deterministic
dynamical systems.
