Caching-enabled device-to-device (D2D) networks have the potential to make mobile users directly fetch requested files from nearby users, resulting in low network delay. In addition, user mobility can increase the communication chances among different users, and therefore, the network delay can be further effectively reduced by proper designing the caching strategy. In this paper, mobility-aware caching strategies in D2D networks are studied to minimize the network delay. Specifically, based on the inter-contact user mobility model, the expression of the average file delivery delay is analytically obtained. Considering the limited cache capacity, a delay minimization cache placement problem considering the user mobility is investigated. To optimally solve this nonlinear integer programming problem, we reformulate it as a multistage decision problem. According to the recursive relationship between adjacent stages, dynamic programming is adopted to obtain the optimal mobility-aware caching strategy stage-by-stage. Furthermore, to lower the complexity, we also demonstrate that the original problem can be recasted as a monotone submodular function maximization problem over a matroid constraint. Then, a low-complexity greedy mobility-aware caching strategy with (1-1/e)-optimality performance guarantee is put forward. Numerical results show that, in the scenario with high user mobility, the file delivery delay can be reduced by 47% with our proposed mobility-aware caching strategy, as compared with the most popular caching. Furthermore, the superiority of the proposed caching strategy is verified by real-world data set.