The spatially dependent description of the electron kinetics in nonuniform electric fields is of primary importance for the modeling of gas-filled proportional counters and other plasma devices. For a typical example, the amplification of the gain signal in the gas counters is determined by the behavior of electron ionization processes in the cylindrically or spherically symmetric electric field around thin wire or tiny sphere anode. In this paper, the general formalism of the Boltzmann equation in these types of the nonuniform electric fields is presented for specifying the electron swarm dynamics affected by the field geometry. The behavior of electrons in the cylindrical or spherical field configuration is investigated by a Monte Carlo technique to exemplify the description, and the effects associated with the angular momentum of electrons are discussed in terms of the ionization coefficient.