In this paper we further develop the formulation of the method of tunneling currents for the description of the tunneling transition in long-distance bridge-mediated electron transfer reactions introduced in our previous work [A. A. Stuchebrukhov, J. Chem. Phys. 104, 8424 (1996); 105, 10819 (1996)]. Here we present a full many-election treatment of the problem in the case when the atomic basis set employed for the description of the medium is nonorthogonal. In this formulation we introduce many-electron Mulliken population operator and derive a set of kinetic equations describing evolution of different atomic states during the tunneling transition. The analysis of the kinetic equations naturally leads then to a concept of electron density fluxes, or currents, between atoms of the medium propagating the tunneling electron. Explicit formulas expressing interatomic tunneling currents in terms of the coefficients of expansion of molecular orbitals of donor and acceptor diabatic electronic states in the atomic basis set are derived. Specific effects due many-electron nature of the system and non-orthogonality (overlaps) of the atomic states are discussed.