The breakdown of different air gaps at high overvoltages in an inhomogeneous electric field was investigated with a time resolution of up to 100 ps. Dynamic displacement current was used for diagnostics of ionization processes between the ionization wave front and a plane anode. It is demonstrated that during the generation of a supershort avalanche electron beam (SAEB) with amplitudes of ~10 A and more, conductivity in the air gaps at the breakdown stage is ensured by the ionization wave, whose front propagates from the electrode of small curvature radius, and by the dynamic displacement current between the ionization wave front and the plane electrode. The amplitude of the dynamic displacement current measured by a current shunt is 100 times greater than the SAEB. It is shown that with small gaps and with a large cathode diameter, the amplitude of the dynamic displacement current during a subnanosecond rise time of applied pulse voltage can be higher than 4 kA.