Turbulent dispersion of ion gyrocenters in a magnetized plasma is studied in the context of a stochastic Hamiltonian transport model and nonlinear, self-consistent gyrokinetic simulations. The Hamiltonian model consists of a superposition of drift waves derived from the linearized Hasegawa-Mima equation and a zonal shear flow perpendicular to the density gradient. Finite Larmor radius (FLR) effects are included. Because there is no particle transport in the direction of the density gradient, the focus is on transport parallel to the shear flow. The prescribed flow produces strongly asymmetric non-Gaussian probability distribution functions (PDFs) of particle displacements, as was previously known. For kρ=0, where k is the characteristic wavelength of the flow and ρ is the thermal Larmor radius, a transition is observed in the scaling of the second moment of particle displacements. The transition separates nearly ballistic superdiffusive dispersion from weaker superdiffusion at later times. FLR effects eliminate this transition. Important features of the PDFs of displacements are reproduced accurately with a fractional diffusion model. The gyroaveraged ExB drift dispersion of a sample of tracer ions is also examined in a two-dimensional, nonlinear, self-consistent gyrokinetic particle-in-cell (PIC) simulation. Turbulence in the simulation is driven by a density gradient and magnetic curvature, resulting in the unstable ρ scale kinetic entropy mode. The dependence of dispersion in both the axial and radial directions is characterized by displacement and velocity increment distributions. The strength of the density gradient is varied, using the local approximation, in three separate trials. A filtering procedure is used to separate trajectories according to whether they were caught in an eddy during a set observation time. Axial displacements are compared to results from the Hasegawa-Mima model. Superdiffusion and ballistic transport are found, depending on filtering and strength of the gradient. The radial dispersion of particles, as measured by the variance of tracer displacements, is diffusive. The dependence of the running diffusion coefficient on ρ for each value of the density gradient is considered.