  We investigate quantum dynamics with the underlying Hamiltonian being a
Jacobi or a block Jacobi matrix with the diagonal and the off-diagonal terms
modulated by a periodic or a limit-periodic sequence. In particular, we
investigate the transport exponents. In the periodic case we demonstrate
ballistic transport, while in the limit-periodic case we discuss various
phenomena such as quasi-ballistic transport and weak dynamical localization. We
also present applications to some quantum many body problems. In particular, we
establish for the anisotropic XY chain on $\mathbb{Z}$ with periodic parameters
an explicit strictly positive lower bound for the Lieb-Robinson velocity.
