We investigate thermofluidic transport phenomena and entropy generation for combined electroosmotic and pressure-driven flows through narrow confinements, subjected to spatially varying non-uniform magnetic field. Going beyond the Debye-Huckel limit, we consider the size effects of the ionic species (steric effect) to analyse magnetohydrodynamic flow and heat transfer characteristics. We demonstrate 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. The novel scope of the current research lies in the state-of-art design of advanced micromechanical industrial smart-sensors, actuators, and biomedical devices. (C) 2016 Elsevier Ltd. All rights reserved.