Magnetic vortices are characterized by the sense of in-plane magnetization circulation and by the polarity of the vortex core. With each having two possible states, there are four possible stable magnetization configurations that can be utilized for a multibit memory cell. Dynamic control of vortex core polarity has been demonstrated using both alternating and pulsed magnetic fields and currents. Here, we show controlled dynamic switching of spin circulation in vortices using nanosecond field pulses by imaging the process with full-field soft X-ray transmission microscopy. The dynamic reversal process is controlled by far-from-equilibrium gyrotropic precession of the vortex core, and the reversal is achieved at significantly reduced field amplitudes when compared with static switching. We further show that both the field pulse amplitude and duration required for efficient circulation reversal can be controlled by appropriate selection of the disk geometry.