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Electrical breakdown of heterogeneous dielectrics in the form of rock samples was studied. The studies were conducted in an inhomogeneous field using the scheme conical tip - plane. The high-voltage generator of 10 ns pulses duration with a slope of 3 ns rise and peak voltage up to 300 kV was employed. Forms of incident and reflected voltage pulses were recorded by a coaxial meter, which allowed characterizing breakdown dynamical parameters of the samples. A model of development of the breakdown based on electrostrictive mechanism was proposed. The dynamics of motion of electrons and ions in the micro-cracks formed by high-voltage pulses with nanosecond fronts near impurities in the materials with developed impurity phase was investigated numerically. Characteristic times of microgap crossing and maximum electron energy were determined. For the breakdown field, maximum electron energy can reach tens of electron volts, which causes ionization of atoms of the substance including deep levels. The role of space charge in the electron dynamics was also studied.