We apply the nonequilibrium functional renormalization group approach treating flow of the electronic self-energies, to describe local magnetic moment formation and electronic transport in a quadruple quantum dot (QQD) ring, coupled to leads, with moderate Coulomb interaction on the quantum dots. We find that at zero temperature depending on parameters of the QQD system, the regimes with zero, one, or two almost local magnetic moments in the ring can be realized, and the results of the considered approach in equilibrium agree qualitatively with those of a more sophisticated functional renormalization group approach treating also flow of the vertices. It is shown that the almost formed local magnetic moments, which exist in the equilibrium, remain stable in a wide range of bias voltages near equilibrium. The destruction of the local magnetic moments with increasing bias voltage is realized in one or two stages, depending on the parameters of the system; for the two-stage process the intermediate phase possesses fractional magnetic moments. We present zero-temperature results for current-voltage dependencies and differential conductances of the system, which exhibit sharp features at the transition points between different magnetic states. The occurrence of the interaction-induced negative differential conductance phenomenon is demonstrated and discussed. For one local moment in the ring and finite hopping between the opposite quantum dots, connected to the leads, we find suppression of the conductance for one of the spin projections in infinitesimally small magnetic field, which occurs due to destructive interference of different electron propagation paths and can be used in spintronic devices.