Flexible metal–organic frameworks (MOFs) undergo reversible structural transformations triggered by external stimuli. An interesting feature of some MOFs is their ability to flex in response to specific guests, which can enable selective separation. Here, we introduce MUF-15-OMe ([Co6(μ3-OH)2(ipa-OMe)5(H2O)4]), a variant of MUF-15 that comprises hexanuclear cobalt(II) clusters connected by 5-methoxyisophthalate (ipa-OMe) ligands. MUF-15 itself has isophthalate linkers and is inflexible upon uptake of common gases. On the other hand, MUF-15-OMe flexes upon the uptake of gases such as CO2 and C2 hydrocarbons at pressures less than 1 bar, as revealed by distinct steps in its gas adsorption isotherms. Computational analysis showed that the underlying mechanism involves partial detachment of one of the carboxyl groups of the framework linkers. The gas pressure required to induce framework dynamics can be tuned by replacing some of the ipa-OMe by isophthalate ligands in multivariate frameworks. The flexing of MUF-15-OMe opens up space for the adsorption of specific additional gas molecules. This enhances the separation of CO2 and N2 and enables the differentiation of H2 and D2 by quantum sieving. By providing a clear illustration of how flexibility allows the discrimination of gas mixtures, this study underpins the use of dynamic MOFs for challenging separations.