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The temporal characteristics of the Xe$^+$ ion axial Velocity Distribution Function (VDF) were recorded in the course of low-frequency discharge current oscillations ($\sim$~14 kHz) of the 5 kW-class PPS$\circledR$X000 Hall thruster. The evolution in time of the ion axial velocity component is monitored by means of a laser induced fluorescence diagnostic tool with a time resolution of 100 ns. As the number of fluorescence photons is very low during such a short time period, a hom-made pulse-counting lock-in system was used to perform real-time discrimination between background photons and fluorescence photons. The evolution in time of the ion VDF was observed at three locations along the thruster channel axis after a fast shut down of the thruster power. The anode discharge current is switched off at 2 kHz during 5 $\mu$s without any synchronization with the current oscillation cycle. This approach allows to examine the temporal behavior of the ion VDF during decay and ignition of the discharge as well as during forced and natural plasma oscillations. Measurements show that the distribution function of the axial component of the Xe$^+$ ion does change periodically in time with a frequency close to the current oscillation frequency in both forced and natural cases. The ion density and the mean velocity are found to oscillate whereas the velocity dispersion stays constant, which indicates that ionization and acceleration layers have identical dynamics. Finally, variations over time of the electric field are for the first time experimentally evidenced in a crossed-field discharge., Comment: 21 pages, 10 figures