The Internet of Things (IoT) is ubiquitous thanks to the rapid development of wireless technologies. However, the broadcast nature of wireless transmissions results in great vulnerability to device authentication. Physical layer authentication emerges as a promising approach by exploiting the unique channel characteristics. However, a practical scheme applicable to dynamic channel variations is still missing. In this paper, we proposed a deep learning-based physical layer channel state information (CSI) authentication for mobile scenarios and carried out comprehensive simulation and experimental evaluation using IEEE 802.11n. Specifically, a synthetic training dataset was generated based on the WLAN TGn channel model and the autocorrelation and the distance correlation of the channel, which can significantly reduce the overhead of manually collecting experimental datasets. A convolutional neural network (CNN)-based Siamese network was exploited to learn the temporal and spatial correlation between the CSI pair and output a score to measure their similarity. We adopted a synergistic methodology involving both simulation and experimental evaluation. The experimental testbed consisted of WiFi IoT development kits and a few typical scenarios were specifically considered. Both simulation and experimental evaluation demonstrated excellent generalization performance of our proposed deep learning-based approach and excellent authentication performance. Demonstrated by our practical measurement results, our proposed scheme improved the area under the curve (AUC) by 0.03 compared to the fully connected network-based (FCN-based) Siamese model and by 0.06 compared to the correlation-based benchmark algorithm.