We present a new model of time-dependent flow at low latitudes in the Earth's core between 2000 and 2018, derived from magnetic field measurements made on board the {\it Swarm} and CHAMP satellites and at ground magnetic observatories. The model, called {\it CoreFlo-LL.1}, consists of a steady background flow without imposed symmetry plus a time-dependent flow that is dominated by geostrophic and quasi-geostrophic components but also allows weak departures from equatorial symmetry. We find that the equatorial region beneath the core-mantle boundary is a place of vigorous, localised, fluid motions; time-dependent flow focused at low latitudes close to the core surface is able to reproduce rapid field variations observed at non-polar latitudes at and above Earth's surface. Magnetic field acceleration pulses are produced by alternating bursts of non-zonal azimuthal flow acceleration in this region. Such acceleration sign changes can occur within a year or less, and when the structures involved are of large spatial scale they can give rise to geomagnetic jerks at the Earth's surface.