Content caching in the edge of wireless networks is a promising technology to reduce the backhaul traffic of duplicated data transmission. Its key issue lies in the accurate prediction of user requirements. Considering the pervasive movement of users in cellular networks, especially in small-cell networks, we propose a mobility prediction-based content caching replacement strategy in this paper. Note that the impact of unevenly distributed file popularity is much larger in small cells than in macro cells due to their small coverage, where the local file request profile does not match the global one. In more detail, the user location predicted by long short term memory (LSTM) is incorporated into the caching replacement algorithm based on a deep reinforcement learning (DRL) framework. Simulation results show that the mobility prediction brings significant performance improvement in terms of cache hit ratio (CHR) in various movement scenarios, especially for a more regular movement pattern of users. Moreover, the optimal CHR threshold in the proposed algorithm is analytically derived, and the performance impact of learning rate as well as the storage size is also well investigated.