The answers in this post suggest charges move through the bulk of conductors, not only their surface, when dealing with direct currents.
However, suppose we have a steady direct current, this steady state condition implies $\nabla .\textbf{J}=0$ within the conductor. Furthermore, conductors, by definition, satisfy $\,\textbf{J}=\sigma \textbf{E}$. From these two equations, we find $$\nabla. \textbf{E} =0$$
again, within the conductor. And since $\nabla. \textbf{E} =\frac{\rho}{\epsilon _{0}}$, we will ultimately have $$\rho =0$$
inside the conductor. This is in clear contradiction with the statement that charges move inside conductors for direct currents.
P.S. The reasoning above can be found in Purcell's Electricity and Magnetism, page 131.
