In this study, a polyacrylonitrile nanofiber membrane was first hydrolyzed and then functionalized with tris(hydroxymethyl)aminomethane (P-Tris), then used as an affinity nanofiber membrane for lysozyme adsorption in membrane chromatography. The dynamic adsorption behavior of lysozyme was investigated in a flow system under various operating parameters, including adsorption pHs, initial feed lysozyme concentration, loading flow rate, and the number of stacked membrane layers. Four different kinetic models, pseudo-first-order, pseudo-second-order, Elovich, and intraparticle diffusion kinetic models, were applied to experimental data from breakthrough curves of lysozyme. The results showed that the dynamic adsorption results were fitted well with the pseudo-second-order kinetic model. The breakthrough curve experimental results show significant differences in the breakthrough time, the dynamic binding capacity, the length of the mass transfer zone, and the utilization rate of the membrane bed under different operating parameters. Four dynamic adsorption models (i.e., Bohart–Adams, Thomas, Yoon–Nelson, and BDST models) were used to analyze the breakthrough curve characteristics of the dynamic adsorption experiments. Among them, the Yoon–Nelson model was the best model to fit the breakthrough curve. However, some of the theoretical results based on the Thomas and Bohart–Adams model analyses of the breakthrough curve fit well with the experimental data, with an error percentage of <5%. The Bohart–Adams model has the largest difference from the experimental results; hence it is not suitable for breakthrough curve analysis. These results significantly impact dynamic kinetics studies and breakthrough curve characteristic analysis in membrane bed chromatography.