Note that there are two types of electrochemical cells: galvanic, also called Voltaic, and electrolytic. Galvanic/Voltaic cells derive energy from spontaneous redox reactions, while electrolytic cells involve non-spontaneous reactions and thus require an external electron source like a DC battery or an AC power source. Ref 1.
If the cell generates its own voltage (i.e, we are looking at a galvanic/Voltaic cell), when the reactants get used up, current flow through an external circuit will gradually decrease to zero, or an immeasurably small amount. This is the case in a typical 1.5V battery (or a 3 V mercury cell). But you are surely familiar with the situation of batteries getting used up, either partially or completely.
The conduction of electricity may be stopped in an electrolytic cell if you disconnect the power source from the cell completely, or if you reduce the voltage applied to the cell to such a low level that the anions and cations cannot discharge.
Perhaps you wonder if electrolyzing a solution of NaCl uses up the NaCl and could bring about a complete stop of current flow. No, not really, but things do change. If you could completely flush out the chlorine gas as soon as it is produced, the sodium ion will still remain. If Na$^+$ were reduced to metal, it would react immediately to give H2, but that doesn't even happen (although, if you used a mercury cathode, you could get sodium amalgam, which would slowly decompose to Na$^+$ and H2). The Na$^+$ just carries 4 or 5 or 6 H2O molecules close to the cathode where an electron turns a water molecule into a proton and a hydroxide ion. Eventually, all the NaCl could be converted to NaOH - actually, the conductivity would increase! I suppose you could make the current go to zero by electrolyzing all the water away, to O2 and H2, but the electrolyte would still work if you add more water.
Similar situations occur with most of the electrolytes you will encounter, like NaCl, KCl, CuSO4, Zn(NO3)2, etc.
Now it is conceivable that some electrolytes could degrade by leaving the solution, as in the case of chloride ions. Organic acids could be degraded to paraffins and CO2; nitrates could be flushed out as nitrogen oxides, noble metals (Cu, Ag, Au, Pt) could plate out, but carrying electrons to or from electrodes does not require that the ions die or become ineffective.
Ref 1. https://www.khanacademy.org/test-prep/mcat/physical-processes/intro-electrochemistry-mcat/a/electrochemistry