This paper investigates the impact of mobility on underwater acoustic communication networks in which the propagation delay is comparable to or larger than the packet duration. An underwater acoustic wireless network, consisting of static and mobile nodes, is studied for its link-layer channel utilization. Synchronous and asynchronous media access control (MAC) protocols are employed with ALOHA, TDMA (time-division multiple access), and artificial intelligence (AI) agent nodes. The simulation results of a multi-node network show that the asynchronous MAC protocols achieve up to 6.66× higher channel utilization than synchronous protocols by allowing time slots to be shorter than the maximum propagation delay among nodes and permitting asynchronous transmission time. The high mobility of a few mobile nodes also favors asynchronous protocols and increases the overall channel utilization. However, node mobility causes more difficulties for the AI node to learn the environment, which may be ineffective to achieve higher gains in channel utilization.