We report an abrupt change in the diffusive transport of inertial objects in wave-driven turbulence as a function of the object size. In these non-equilibrium two-dimensional flows, the turbulent diffusion coefficient $D$ of finite-size objects undergoes a sharp change for values of the object size $r_{p}$ close to the flow forcing scale $L_{f}$. For objects larger than the forcing scale ($r_{p}>L_{f}$), the diffusion coefficient is proportional to the flow energy $U^{2}$ and inversely proportional to the size $r_{p}$. This behaviour, $D\sim U^{2}/r_{p}$ , observed in a chaotic macroscopic system is reminiscent of a fluctuation–dissipation relation. In contrast, the diffusion coefficient of smaller objects ($r_{p}<L_{f}$) follows $D\sim U/r_{p}^{0.35}$. This result does not allow simple analogies to be drawn but instead it reflects strong coupling of the small objects with the fabric and memory of the out-of-equilibrium flow. In these turbulent flows, the flow structure is dominated by transient but long-living bundles of fluid particle trajectories executing random walk. The characteristic widths of the bundles are close to $L_{f}$. We propose a simple phenomenology in which large objects interact with many bundles. This interaction with many degrees of freedom is the source of the fluctuation–dissipation-like relation. In contrast, smaller objects are advected within coherent bundles, resulting in diffusion properties closely related to those of fluid tracers.