  We introduce and study a family of Markov processes on partitions. The
processes preserve the so-called z-measures on partitions previously studied in
connection with harmonic analysis on the infinite symmetric group. We show that
the dynamical correlation functions of these processes have determinantal
structure and we explicitly compute their correlation kernels. We also compute
the scaling limits of the kernels in two different regimes. The limit kernels
describe the asymptotic behavior of large rows and columns of the corresponding
random Young diagrams, and the behavior of the Young diagrams near the
diagonal.
  Our results show that recently discovered analogy between random partitions
arising in representation theory and spectra of random matrices extends to the
associated time-dependent models.
