We numerically investigate the dynamics of two immiscible conductive fluids in a narrow fluidic channel under the combined influence of electric and magnetic fields using a diffuse interface based phase-field model. The numerical solver is validated from two different perspectives, viz., with the reported results of microscale multiphase transport as well as the available experimental results in the paradigm of electrically actuated transport. The magnetic field induces the Lorentz force due to its interaction with the electrical forcing, which in turn leads to complex interfacial dynamics and development of a finger-like interface front of the advancing fluid into the receding fluid. Under certain conditions studied in the present work, the trend reverses, and a finger of receding fluid is formed into the advancing fluid. The effect of contrast in fluid properties is studied and the interface breaking phenomenon is observed beyond a threshold viscosity contrast. It is found that for a given viscosity contrast between the fluids, an increase in the strength of the applied magnetic field prevents wetting failure.