In the transition to a clean-energy future, CO₂ separations will play a critical role in mitigating current greenhouse gas emissions and facilitating conversion to cleaner-burning and renewable fuels. New materials with high selectivities for CO₂ adsorption, large CO₂ removal capacities, and low regeneration energies are needed to achieve these separations efficiently at scale. Here, we present a detailed investigation of nine diamine-appended variants of the metal-organic framework Mg₂(dobpdc) (dobpdc^4- = 4,4'-dioxidobiphenyl-3,3'-dicarboxylate) that feature step-shaped CO₂ adsorption isotherms resulting from cooperative and reversible insertion of CO₂ into metal-amine bonds to form ammonium carbamate chains. Small modifications to the diamine structure are found to shift the threshold pressure for cooperative CO₂ adsorption by over 4 orders of magnitude at a given temperature, and the observed trends are rationalized on the basis of crystal structures of the isostructural zinc frameworks obtained from in situ single-crystal X-ray diffraction experiments. The structure-activity relationships derived from these results can be leveraged to tailor adsorbents to the conditions of a given CO₂ separation process. The unparalleled versatility of these materials, coupled with their high CO₂ capacities and low projected energy costs, highlights their potential as next-generation adsorbents for a wide array of CO₂ separations.