Multiphase transport phenomena primarily encompass the fundamental principles and applications concerning the systems where overall dynamics are precept by phase change evolution. On the other hand, multiscale transport phenomena essentially corroborate to a domain where the transport characteristics often contain components at disparate scales. Relevant examples as appropriate to multiphase and multiscale thermofluidic transport phenomena comprise solid-liquid phase change during conventional solidification process and hydrodynamics through narrow confinements. The additional effect of superimposed magnetic field over such multiphase and multiscale systems may give rise to intriguing transport characteristics, significantly unique in nature as compared to flows without it. The present investigation focuses on multiphase, multi-scale transport phenomena in physical systems subjected to the superimposed magnetic field, considering four important and inter-linked aspects. To begin with, for a multiphase system concerning binary alloy solidification, a normal mode linear stability analysis has been carried out to investigate stationary and oscillatory convective stability in the mushy layer in the presence of external magnetic field. The stability results indicate that the critical Rayleigh number for stationary convection shows a linear relationship with increasing Ham (mush Hartmann number). Magnetohydrodynamic effect imparts a stabilizing influence during stationary convection. In comparison to that of stationary convective mode, the oscillatory mode appears to be critically susceptible at higher values of  (a function of the Stefan number and concentration ratio), and vice versa for lower  values. Analogous to the behaviour for stationary convection, the magnetic field also offers a stabilizing effect in oscillatory convection and thus influences global stability of the mushy layer. Increasing magnetic strength shows reduction in the wavenumber and in the number of rolls formed in the mushy layer. In multiscale paradigm, the combined electroosmotic and pressure-driven transport through narrow confinements have been firstly analyzed with an effect of spatially varying non–uniform magnetic field. It has been found that a confluence of the steric interactions with the degree of wall charging (zeta potential) may result in heat transfer enhancement, and overall reduction in entropy generation of the system under appropriate conditions. In particular, it is revealed that a judicious selection of spatially varying magnetic field strength may lead to an augmentation in the heat transfer rate. It is also inferred that incorporating non–uniformity in distribution of the applied magnetic field translates the system to be dominated by the heat transfer irreversibility. Proceeding further, a semi-analytical investigation has been carried out considering implications of magnetohydrodynamic forces and interfacial slip on the heat transfer characteristics of streaming potential mediated flow in narrow fluidic confinements. An augmentation in the streaming potential field as attributable to the wall slip activated enhanced electromagnetohydrodynamic transport of the ionic species within the EDL has been found. Furthermore, the implications of Stern layer conductivity and magnetohydrodynamic influence on system irreversibility have been shown through analysis of entropy generation due to fluid friction and heat transfer. The results being obtained in this analysis have significant scientific and technological consequences in the context of novel design of future generation energy efficient devices, and can be useful in the further advancement of theory, simulation, and experimental work. Finally, the combined consequences of interfacial electrokinetics, rheology, and superimposed magnetic field subjected to a non-Newtonian (power-law obeying) fluid in a narrow confinement are studied in this work. The theoretical results demonstrate that the applied magnetic field imparts a retarding influence in the induced streaming potential development, whereas, triggers the heat transfer magnitude. Moreover, additional influences of power law index show reduction in heat transfer as well as the streaming potential magnitude. It is unveiled that the optimal combinations of power law index and the magnetic field lead to the minimization of the global total entropy generation in the system.