The coalescence of two different drops, one surfactant-laden and the other surfactant-free, was studied under the condition of confined flow in a microchannel. The coalescence was accompanied by penetration of the surfactant-free drop into the surfactant-laden drop because of the difference in the capillary pressure and Marangoni flows causing a film of surfactant-laden liquid to spread over the surfactant-free drop. The penetration rate was dependent on the drop order, with considerably better penetration observed for the case when the surfactant-laden drop goes first. The penetration rate was found to increase with an increase of interfacial tension difference between the two drops, an increase of flow rate and drop confinement in the channel (for the case of the surfactant-laden drop going first), an increase of viscosity of the continuous phase, and a decrease of viscosity of the dispersed phase. Analysis of flow patterns inside the coalescing drops has shown that, unlike coalescence of identical drops, only two vortices are formed by asymmetrical coalescence, centered inside the surfactant-free drop. The vortices were accelerated by the flow of the continuous phase if the surfactant-laden drop preceded the surfactant-free one, increasing the rate of penetration; the opposite was observed if the drop order was reversed. The mixing patterns on a longer time scale were also dependent on the drop order, with better mixing being observed for the case when the surfactant-laden drop goes first.