We investigated the magneto-electrochemical characteristics of two modes of conventional flow-electrode capacitive deionization (FCDI) cells by monitoring Faraday-Hall signals (VFH), conductivity (sigma), and pH levels. In the open-circuit mode, VFH (proportional to v -> x B ->; v ->: fluid velocity, B ->: magnetic field) is additive, symmetric, and weakly dependent on concentration. We observed magnetically-driven ion permeation across the ion-exchange membranes (IEMs), indicating a magnetodialysis process. In the closed-circuit mode, the magnetically-induced potential is approximately 400 times smaller than the deionizing electric potential. Repositioning the current collectors closer to the IEMs within the flow channel drastically reduces internal resistance and eliminates the need for flowable electrodes. For the closed-circuit mode, we discuss the evolution of conductivity, pH, and performance metrics when varying the applied current, magnetic field, and solution concentration. These findings on magnetodialysis and the advantageous repositioning of membrane-and-electrode assemblies in the FCDI cell suggest promising technological prospects, highlighting the need for further refinement, optimization, and scaling up of this approach. In particular, although magnetodialysis is weak in our current setup, it could be enhanced by optimizing: (i) cross product v -> x B ->, (ii) permeability of IEMs, and (iii) role of flowable electrodes in capturing permeated charges and electrode regeneration.