We know that pure water does not conduct electricity, but salt water is a decent conductor. This is commonly explained by saying that “the ions carry the current through the solution”, which is an incomplete explanation, because it does not indicate what happens when all of the ions have migrated to the electrodes.
More complete explanations of conduction through a salty solution (like this one or this one) rationalize the conduction of electricity in terms of a reduction reaction that takes place at the anode and an oxidation reaction that takes place at the cathode. In the case of salt water, chlorine gas ($\ce{Cl2}$) is formed at the anode and hydrogen gas ($\ce{H2}$) is formed at the cathode.
This explanation seems reasonable, but it implies that the conduction of electricity through a solution is fundamentally different than the conduction of electricity through a wire. A copper wire is (usually) unchanged, even after a large amount of electricity passes through it. In contrast, when electricity passes through salt water, two chemical reactions occur (one at each electrode), which fundamentally change the composition material.
This implies that it is not possible for an aqueous solution to conduct electricity forever. Since we are driving a chemical reaction, we are either consuming our salt (forming $\ce{Cl2}$ gas in the case of an NaCl solution or plating it onto the electrodes in other cases), or we are consuming the water by forming $\ce{H2}$ or $\ce{O2}$ gas.
This is surprising to me. So, I’m asking if my thinking is correct: is it possible for an aqueous salt-containing solution to conduct electricity forever, or it will it always eventually consume the reactants and stop conducting electricity as I have surmised?