With the aim of developing more efficient H2 purification adsorbents, we demonstrated the application of novel hybrid nanocomposites comprising of zeolite 5A and MOF-74 with core-shell structure in purification of H2 from SMR off-gas streams, in the first part of this investigation [14]. Through equilibrium adsorption measurements, it was shown that zeolite-5A@MOF-74 with weight ratio of 5:95 exhibited 20–30% increase in CO2, CO, CH4, and N2 uptake than the bare MOF, as a result of its higher surface area and pore volume. In this work, dynamic adsorption performance of zeolite-5A@MOF-74 in H2 purification process was evaluated through binary and multicomponent breakthrough measurements at various pressures. Moreover, high-pressure adsorption isotherms (up to 20 bar) were used to estimate the theoretical selectivity values for (CO2+CO+CH4)/H2 for comparison with actual selectivities estimated from breakthrough profiles. At 20 bar and room temperature, the composite exhibited equilibrium capacities of 13.8, 8.0, 7.7, and 6.7 mmol/g for CO2, CO, CH4, and H2, respectively which were higher than those of the parent adsorbents. Moreover, multicomponent breakthrough results showed that the selectivity for (CO2 + CO + CH4)/H2 over the composite adsorbent is higher than that over the parent MOF-74 and zeolite 5A materials across all pressures. The total mass transfer coefficients estimated from breakthrough simulations ranged from 6.22 × 10−2, 4.73 × 10−2, and 3.29 × 10−2 s−1 for H2 to 9.23 × 10−5, 7.6 × 10−5, and 6.61 × 10−5 s−1 for CO2, for zeolite-5A@MOF-74, MOF-74, and zeolite 5A, respectively at 1 bar, with the coefficients slightly decreasing as total pressure increased to 15 bar.