  We consider a random process with discrete time formed by singular values of
products of truncations of Haar distributed unitary matrices. We show that this
process can be understood as a scaling limit of the Schur process, which gives
determinantal formulas for (dynamical) correlation functions and a contour
integral representation for the correlation kernel. The relation with the Schur
processes implies that the continuous limit of marginals for q-distributed
plane partitions coincides with the joint law of singular values for products
of truncations of Haar-distributed random unitary matrices. We provide
structural reasons for this coincidence that may also extend to other classes
of random matrices.
