Liquid-liquid slug flow regime is characterised by the presence of strong internal circulation which enhances mass transfer. In this work, we develop a mathematical model for conjugate mass transfer in the liquid-liquid slug flow regime in a microchannel. A Lagrangian approach is adopted where the behaviour of a unit cell in the channel is analysed in a moving reference frame. The system is analysed for two cases when (I) the slug is in direct contact with the channel wall and (II) there is a thin film of continuous fluid surrounding the slug. A novel contribution of this work is the extension of the stream function formulation to determine the flow structures in multiply connected domains made up of two liquids in a pressure driven flow. The primary objective of the work is to understand the influence of the flow patterns (which is determined by fluid properties and operating conditions) on the mass transfer of a solute from the slug to the continuous phase. Towards this, the species transport equation is solved numerically using the velocity field obtained. The reliability of the model developed is validated with experiments reported in the literature. This study gives us insights on the influence of the film on the hydrodynamics and its contribution to mass transfer. A key finding of this study is that the rate of mass transfer can be enhanced if the continuous phase has higher viscosity than the slug phase.