Schur dynamics of the Schur processes

We construct discrete time Markov chains that preserve the class of Schur processes on partitions and signatures. One application is a simple exact sampling algorithm for q^(volume)-distributed skew plane partitions with an arbitrary back wall. Another application is a construction of Markov chains on infinite Gelfand–Tsetlin schemes that represent deterministic flows on the space of extreme characters of the infinite-dimensional unitary group