  We introduce a class of integrable dynamical systems of interacting classical
matrix-valued fields propagating on a discrete space-time lattice, realized as
many-body circuits built from elementary symplectic two-body maps. The models
provide an efficient integrable Trotterization of non-relativistic
$\sigma$-models with complex Grassmannian manifolds as target spaces,
including, as special cases, the higher-rank analogues of the Landau-Lifshitz
field theory on complex projective spaces. As an application, we study
transport of Noether charges in canonical local equilibrium states. We find a
clear signature of superdiffusive behavior in the Kardar-Parisi-Zhang
universality class, irrespectively of the chosen underlying global unitary
symmetry group and the quotient structure of the compact phase space, providing
a strong indication of superuniversal physics.
