Radio images of protoplanetary disks demonstrate that dust grains tend to organize themselves into rings. These rings may be a consequence of dust trapping within gas pressure maxima, wherein the local high dust-to-gas ratio is expected to trigger the formation of planetesimals and eventually planets. We revisit the behavior of dust near gas pressure perturbations enforced by a planet in two-dimensional, shearing-box simulations. While dust grains collect into generally long-lived rings, particles with a small Stokes parameter τ s< 0.1 tend to advect out of the ring within a few drift timescales. Scaled to the properties of ALMA disks, we find that rings composed of larger particles (τ s≥ 0.1) can nucleate a dust clump massive enough to trigger pebble accretion, which proceeds to ingest the entire dust ring well within∼ 1 Myr. To ensure the survival of the dust rings, we favor a nonplanetary origin and typical grain size τ s≲ 0.05–0.1. Planet-driven rings may still be possible but if so we would expect the orbital distance of the dust rings to be larger for older systems.