It is believed that the running (for instance, COHERENT) and forthcoming terrestrial neutrino experiments will be sensitive to the neutrino charge radius that is one of the neutrino fundamental electromagnetic characteristics predicted to be non-zero even in the Standard Model. In this work we continue our studies on neutrino oscillations in external environments with large electric currents and constant magnetic fields with account for the diagonal and non-diagonal neutrino charge radii and anapole moments. We derive the corresponding neutrino evolution Hamiltonians in moving matter with electric currents and magnetic fields and calculate the probabilities of the neutrino spin and spin-flavor oscillations. It is explicitly shown that whereas the spin oscillations can be generated by the neutrino anapole moment interactions with an external electric current, the interaction due to the charge radius does not produce the spin oscillations. Thus, these peculiarities can be used for disintegration of the anapole moment and charge radius effects in neutrino interactions.