The use of reactive electrochemical membranes (REM) in flow-through mode during the anodic oxidation of organic compounds makes it possible to overcome the limitations of plate anodes: in the case of REM, the area of the electrochemically active surface is several orders of magnitude larger, and the delivery of organic compounds to the reaction zone is controlled by convective flow rather than diffusion. The main problem with REM is the formation of fouling and gas bubbles in the pores, which leads to a decrease in the efficiency of the process because the hydraulic resistance increases and the electrochemically active surface is shielded. This work aims to study the processes underlying the reduction in the efficiency of anodic oxidation, and in particular the formation of gas bubbles and the recharge of the REM pore surface at a current density exceeding the limiting kinetic value. We propose a simple one-dimensional non-stationary model of the transport of diluted species during the anodic oxidation of paracetamol using REM to describe the above effects. The processing of the experimental data was carried out. It was found that the absolute value of the zeta potential of the pore surface decreases with time, which leads to a decrease in the permeate flux due to a reduction in the electroosmotic flow. It was shown that in the solution that does not contain organic components, gas bubbles form faster and occupy a larger pore fraction than in the case of the presence of paracetamol; with an increase in the paracetamol concentration, the gas fraction decreases. This behavior is due to a decrease in the generation of oxygen during the recombination reaction of the hydroxyl radicals, which are consumed in the oxidation reaction of the organic compounds. Because the presence of bubbles increases the hydraulic resistance, the residence time of paracetamol—and consequently its degradation degree—increases, but the productivity goes down. The model has predictive power and, after simple calibration, can be used to predict the performance of REM anodic oxidation systems.