We carry out a comprehensive series of experiments to understand the effect of multiple-contact miscibility on drainage from a porous medium. Our experiments are performed using a microfluidic porous medium consisting of a random distribution of cylindrical pillars to create a two-dimensional heterogeneous network of pores. A mixture of ethanol and 1-hexanol is injected into the porous medium as a wetting reservoir phase. A mixture of water and ethanol is subsequently injected to displace the reservoir phase. Depending on the precise composition of these two phases, they can be immiscible, develop miscibility over time through mass transfer between them, or be immediately miscible. We describe the distinct behaviors observed depending on the composition and injection flow rate and compare and contrast them with the classical parameter regime diagram for drainage displacement of R. Lenormand et al. [J. Fluid Mech. 189, 165 (1988)]. When the system is immiscible, the injection rate is the dominant control and the behavior is broadly classical. In systems that develop miscibility for the compositions we explore, the fluids remain immiscible at the injection tip and develop miscibility further upstream. Trapped regions of reservoir fluid are mobilized over time through the development of miscibility. For systems that nearly develop miscibility, trapped regions are mobilized at sufficiently high flow rates. We also show that the porosity has limited impact on the behavior and phase-saturation level. Our work has applications to carbon-dioxide enhanced oil recovery.