Amine-functionalized metal organic frameworks (MOFs) such as Mg-2(dobpdc) have shown some of the highest capacities for CO, capture under simulated direct air capture (DAC) conditions, i.e., at a CO2 partial pressure at 0.4 mbar. The most widely studied materials of this kind have adsorption isotherms whose shape is controlled by the material's underlying cooperative adsorption mechanism for CO2. Previous analyses of these materials, however, have focused primarily on equilibrium adsorption. Here, we study adsorption of CO2 in amine-functionalized MOFs under ultradilute conditions relevant to DAC using a packed bed breakthrough system to understand the challenges of using these materials in practical separations. By examining the effects of flow rate, temperature, and CO2 concentration on the observed breakthrough profiles, we show how the isotherm shape, mass transfer resistances and adsorption/desorption kinetics impact the system's performance. An empirical model for the packed bed that accounts for the cooperative reaction mechanism for adsorption of CO, in N,N'-dimethyl ethylene diamine (MMEN)-Mg-2(dobpdc) is developed. The kinetics of CO2 adsorption at CO2 concentrations of 400 ppm are characterized through an Avrami model, while at higher CO2 concentrations, a kinetic model based on a Michaelis-Menten expression is more successful. It is also suggested that, at low concentrations, mass transfer rate is limited by cooperative insertion of CO2. Overall, this work has identified the system performance (i.e., rate of CO2 extraction, CO2 capture fraction, and breakthrough curve types) for different feed concentrations of CO2. Critically, this work shows that the use of materials with stepped isotherm for applications such as DAC has some fundamental challenges that cannot be addressed by efforts that focus solely on modifying the overall equilibrium adsorption capacity or swing capacity.