[Edit] Not sure why this was closed. The answers there do not answer my question, and are not even correct...
Imagine a charge sitting in space. It causes an electric field everywhere, with magnitude $\propto r_{old}^{-2}$
But now let's say we move the charge a little. This will change the electric field everywhere to be $\propto r_{new}^{-2}$. However, according to Maxwell's other equations (I'm told), this propagation is not instant, but happens with a finite speed $c$.
Let's draw a box that intersects the boundary of this wavefront. One end of the box will be have electric field vectors with magnitude $\propto r_{old}^{-2}$, while the other end will be $\propto r_{new}^{-2}$. These are not equal, so $\nabla \cdot E \neq 0$. But since there's no charge in the box, this should be impossible according to Gauss's law!
What's going on?