Turbulent mixing in the ocean can be characterized through the relative dispersion of pairs of fluid particles. The relation between the observed dispersion behaviors and the statistical properties of the underlying turbulent flow, however, can be subtle and does not seem to be fully understood. To address this point, we numerically study relative dispersion in a class of generalized two-dimensional turbulent flows. The latter includes two systems that are relevant for ocean dynamics, the barotropic quasi-geostrophic (QG) model, corresponding to flows with energy concentrated at mesoscales, and the surface quasi-geostrophic (SQG) one, accounting for energetic submesoscales. All the considered dynamics are characterized by the conservation of an active tracer along the geostrophic flow with a direct cascade of tracer variance to small scales, but they theoretically possess different properties in terms of locality of spectral energy transfers. Aiming at exploring the relation between such Eulerian locality features and the statistical properties of the passively transported Lagrangian tracers, we examine relative dispersion in these flows, both as a function of time and as a function of scale, and compare it to predictions based on phenomenological arguments assuming the locality of the cascade. We find that dispersion behaviors agree with expectations from local theories when the dynamics are close to SQG ones and initial pair separations are small enough. Non-local dispersion is instead observed for the QG model, a robust result when looking at relative displacement probability distributions.