How does conductivity work for non-redoxed ions?
This can arise due to ion-ion interactions brought on by Coulombic fields.

A proper explanation is rooted in transport phenomena. Unfortunately chemists are largely not taught this, but chemical engineers do get the opportunity.

Under standard conditions, ions in a solution will take a random walk during diffusion. This results in no net movement of our ions over some span of time. Such a system perturbed by an applied electric field will cause ions to preferentially move in one direction, we say the ions will drift. We say they have a drift velocity. These velocities are affected by a number of parameters, as there is still continual collision, but we can say that the mobilities will be unequal for ions of unequal sizes.

Different mobilities mean concentration gradients form and charge separation too. Typically we use an electroneutrality field as a reasonable approximation to allow analytic solutions for various equations. This field depends on all the ionic fluxes present and as such will affect the total measurable current.

The result is simple: the introduction of any ion will cause a perturbation in the field that all the ions encounter in solution.