Huh? A moving charge still has a charge. Imagine a charge density described by the function over space and time rho(r,t). Current is defined by the continuity e

Huh? A moving charge still has a charge. Imagine a charge density described by the function over space and time rho(r,t). Current is defined by the continuity equation Del[J] = - d rho(r,t) / d t (those are partial derivatives). Basically that says that the rate of increase of the charge density inside an infinitesimal box is related to the total current flowing into the box, i.e. if the charge in a box changes, it must have flowed in/out from just outside the box, i.e. continuity (assuming no sources/sinks). Both exist at the same time. In SR, you can look at the relativistic four-current. Maybe you're thinking of how a current can flow in a wire, despite the wire having no charge. This is because, on our length scales, the electric field of the positively charged atomic nuclei completely cancels the electric field of the negatively charged electrons.