This work theoretically investigates the interfacial transport of immiscible fluid layers in an inclined fluidic channel in the presence of magnetofluidic actuation. Immiscible binary system consists of both non-Newtonian fluid (top layer) and Newtonian fluid (bottom layer), while the Carreau fluid model is used to describe the rheology of non-Newtonian layer. We develop a theoretical framework consistent with the homotopy analysis method (HAM) to obtain the approximate analytical solutions for the underlying thermofluidic transport features. By depicting the auxiliary parameter curve (-curve) of flow velocity and temperature distribution, we ascertain the effective reliability of the theoretical method developed here. We demonstrate both velocity and temperature variations in the channel for a set of involving parameters pertinent to this analysis. Albeit the flow configuration considered in this analysis is not complex, yet, the method developed here seems to be efficient in capturing underling transport features, retaining the simultaneously acted implications of fluid rheology and magnetohydrodynamics. Form the consistency observed in predicting the flow velocity for any values of shear-thinning parameter, including nonintegers, our semi-analytical method is deemed pertinent to predict the thermohydrodynamics of immiscible multilayer system even by accurately capturing the intervening effects of fluid rheology and applied fields.