  We define a measure on tuples of partitions, called the symplectic Schur
process, that should be regarded as the right analogue of the Schur process of
Okounkov-Reshetikhin for the Cartan type C. The weights of our measure include
factors that are universal symplectic characters, as well as a novel family of
"Down-Up Schur functions" that we define and for which we prove new identities
of Cauchy-Littlewood-type. Our main structural result is that the point process
corresponding to the symplectic Schur process is determinantal and we find an
explicit correlation kernel. We also present dynamics that preserve the family
of symplectic Schur processes and explore an alternative sampling scheme, based
on the Berele insertion algorithm, in a special case. Finally, we study the
asymptotics of the Berele insertion process and find explicit formulas for the
limit shape and fluctuations near the bulk and the edge. One of the limit
regimes leads to a new kernel that resembles the symmetric Pearcey kernel.
