We consider the evolution of a family of two-dimensional (2D) dispersive turbulence models. The members of this family involve the nonlinear advection of a dynamically active scalar field, and as per convention, the locality of the streamfunction-scalar relation is denoted by α, with smaller α implying increased locality (α=1 gives traditional 2D dynamics). The dispersive nature arises via a linear term whose strength, after nondimensionalization, is characterized by a parameter ϵ. Setting 0<ϵ≤1, we investigate the interplay of advection and dispersion for differing degrees of locality. Specifically, we study the forward (inverse) transfer of enstrophy (energy) under large-scale (small-scale) random forcing along with the geometry of the scalar field. Straightforward arguments suggest that for small α the scalar field should consist of progressively larger isotropic eddies, while for large α the scalar field is expected to have a filamentary structure resulting from a stretch and fold mechanism, much like that of a small-scale passive field when advected by a large-scale smooth flow. Confirming this, we proceed to forced/dissipative dispersive numerical experiments under weakly nonlocal to local conditions (i.e., α≤1). In all cases we see the establishment of well-defined spectral scaling regimes. For ϵ∼1, there is quantitative agreement between nondispersive estimates and observed slopes in the inverse energy transfer regime. On the other hand, forward enstrophy transfer regime always yields slopes that are significantly steeper than the corresponding nondispersive estimate. At present resolution, additional simulations show the scaling in the inverse regime to be sensitive to the strength of the dispersive term: specifically, as ϵ decreases, quite expectedly the inertial-range shortens but we also observe that the slope of the power law decreases. On the other hand, for the same range of ϵ values, the forward regime scaling is observed to be fairly universal.