We present an experimental and theoretical study of the breakthrough performance of the flexible metal-organic framework Cu(bpy)2(BF4)2 (bpy = 4,4'-bipyridine), termed ELM-11. Pure CO2, He, CH4, and N2 gases, as well as binary gas mixtures of those species, were used to perform breakthrough experiments on ELM-11. ELM-11 exhibits a stepped breakthrough curve for CO2 not seen in rigid adsorbents. By comparing the step heights observed in the experimental breakthrough curves with predictions of the gate pressure obtained from the osmotic framework adsorbed solution theory (OFAST) method, we show that the OFAST method can be used to predict the occurrence and height of the steps observed in the breakthrough curves of flexible metal-organic frameworks. For specific gas mixtures, breakthrough curves on ELM-11 show a "doorstop"-type effect, wherein the observed step heights for CO2 breakthrough curves are reduced when the gas mixture contains small kinetic diameter gas species such as helium.