Vortices have long been speculated to play a role in planet formation, via the collection of dust in the pressure maxima that arise at the cores of vortices in protoplanetary discs (PPDs). The question remains, however: as dust collects in the core of a vortex, when does that vortex remain stable and able to collect further dust, and when and why does it break up? We study this question by running high-resolution 2D simulations of dust-laden vortices. By using the terminal velocity approximation in a local shearing box, it was possible to efficiently run simulations of back-reacting dust in a gas at high resolution. Our results show how the stability of 2D dust-laden vortices in PPDs depends on their size relative to the disc scale height, as well as the dust coupling. We find small vortices with semiminor axis much smaller than the scale height to be stable for the duration of the simulations (t > 2000 orbits). Larger vortices, with semiminor axis smaller than but of the order of scale height, exhibit a drag instability after undergoing a long period of contraction where the core becomes progressively more dust rich. The lifetime of these vortices depends on the dust size, with larger dust grains causing the instability to occur sooner. For the size ranges tested in this paper, micrometre- to millimetre-sized grains, vortices survived for several hundreds of orbits. The result implies that the stability of vortices formed by vertical shear instability and zombie vortex instability, or the breakup of larger vortices through hydrodynamic instabilities, is affected by the presence of dust in the disc. The lifetimes observed in this paper, while shortened by the presence of dust for larger vortices, were still long enough to lead to considerable dust enrichment in the vortex cores.