Elements for nondiffusive transport have been identified in a plasma turbulence model based on the slab drift-wave model. Motivated by the self-organized criticality paradigm, a standard set of drift-wave equations in doubly-periodic spatial domain has been elevated to include a flux-driven background profile with critical gradients. The profile is maintained by the turbulence induced flux from the source to the sink. Tracers that follow the Lagrangian trajectories are the primary transport characterization technique. The competition between down-gradient relaxations and self-generated flows highlights the dual reactions to local steepening of profile gradients, which leads to different transport regimes. An additional external sheared flow further inhibits down-gradient transfer and acts as another critical threshold condition that can lead to flow-driven instabilities. Superdiffusive transport is observed primarily when radial relaxation events dominate while subdiffusive character become more prominent with self-generated and external poloidal flows. Diffusive transport exists when the superdiffusive and subdiffusive components are in balance. The interplay between turbulent relaxation and self-generated sheared poloidal flows, that form the basis for the transport explored in this model, is absent unless a flux-driven setup is used. Most of the rich dynamics were not present when running the simplified model without an equation for background profile evolution. Nondiffusive transport characteristics can also be recovered from a passive scalar field that is advected by the turbulent flow with an inherent diffusivity. The spread of a highly localized cloud of tracers and a passive scalar field reasserts the equivalence between the Lagrangian and quasi-Lagrangian frames. The coincidence between the passive scalar field with the tracers provide a regime of validity where existing experimental technique can be used to characterize transport from two-dimensional experimental data. The results from this work highlight the key features of flux-driven turbulent transport leading to nondiffusive transport. Specifcally, the dual reactions to the local steepening of profile gradients exposes the multiscale feature of turbulent transport that becomes more apparent under a flux-driven profile. The quantification of nondiffusive transport characteristics from the evolution of a passive scalar can have important implication towards the fundamental understanding of fluid turbulence and turbulent transport.