Context. Near- and mid-infrared interferometric observations have revealed populations of hot and warm dust grains populating the inner regions of extrasolar planetary systems. These are known as exozodiacal dust clouds, or exozodis, reflecting the similarity with the solar system’s zodiacal cloud. Radiative transfer models have constrained the dust to be dominated by tiny submicron-sized, carbon-rich grains that are accumulated very close to the sublimation radius. The origin of this dust is an unsolved issue. Aims. We explore two exozodiacal dust production mechanisms, first re-investigating the Poynting-Robertson drag pile-up scenario, and then elaborating on the less explored but promising exocometary dust delivery scenario. Methods. We developed a new, versatile numerical model that calculates the dust dynamics, with non-orbit-averaged equations for the grains close to the star. The model includes dust sublimation and incorporates a radiative transfer code for direct comparison to the observations. We consider in this study four stellar types, three dust compositions, and we assume a parent belt at 50 au. Results. In the case of the Poynting-Robertson drag pile-up scenario, we find that it is impossible to produce long-lived submicron-sized grains close to the star. The inward drifting grains fill in the region between the parent belt and the sublimation distance, producing an unrealistically strong mid-infrared excess compared to the near-infrared excess. The dust pile-up at the sublimation radius is by far insufficient to boost the near-IR flux of the exozodi to the point where it dominates over the mid-infrared excess. In the case of the exocometary dust delivery scenario, we find that a narrow ring can form close to the sublimation zone, populated with large grains from several tens to several hundreds of micrometers in radius. Although not perfect, this scenario provides a better match to the observations, especially if the grains are carbon-rich. We also find that the number of active exocomets required to sustain the observed dust level is reasonable. Conclusions. We conclude that the hot exozodiacal dust detected by near-infrared interferometry is unlikely to result from inward grain migration by Poynting-Robertson drag from a distant parent belt, but could instead have an exocometary origin.