The presence of rings and gaps in protoplanetary discs are often ascribed to planet-disc interactions, where dust and pebbles are trapped at the edges of planetary induced gas gaps. Recent work has shown that these are likely sites for planetesimal formation via the streaming instability. Given the large amount of planetesimals that potentially form at gap edges, we address the question of their fate and ability to radially transport solids in protoplanetary discs. We perform a series of N-body simulations of planetesimal orbits, taking into account the effect of gas drag and mass loss via ablation. We consider two planetary systems: one akin to the young Solar System, and another one inspired by HL Tau. In both systems, the close proximity to the gap-opening planets results in large orbital excitations, causing the planetesimals to leave their birth locations and spread out across the disc soon after formation. Planetesimals that end up on eccentric orbits interior of 10au experience efficient ablation, and lose all mass before they reach the innermost disc region. In our nominal Solar System simulation with $dot{M}_0=10^{-7}, M_{odot}, textrm{yr}^{-1}$ and $alpha=10^{-2}$, we find that 70% of the initial planetesimal mass has been ablated after 500kyr. The ablation rate in HL Tau is lower, and only 11% of the initial planetesimal mass has been ablated after 1Myr. The ablated material consist of a mixture of solid grains and vaporized ices, where a large fraction of the vaporized ices re-condense to form solid ice. Assuming that the solids grow to pebbles in the disc midplane, this results in a pebble flux of $sim 10-100,M_{oplus}textrm{Myr}^{-1}$ through the inner disc. Our results demonstrate that scattered planetesimals can carry a significant flux of solids past planetary-induced gaps in young and massive protoplanetary discs.
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