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We consider the energy-efficient mobility-aware content caching problem with clustered small cells (SCs) in ultra-dense networks (UDNs). The UDN is regarded as one of the key solutions to overcome the explosive growth of mobile data traffic and consequential energy consumption. In addition, caching the popular contents at the edge of UDNs can further mitigate the challenges of traffic and energy. In this paper, we group the SCs into disjoint clusters and model the mobility of users between clusters as a Markov chain. We formulate the caching problem aimed at minimizing the overall energy consumption for content delivery to the moving users. Then we decompose the complicated task of energy consumption optimization in cache-enabled UDNs into two sub-problems. In the first sub-problem, we formulate the inter-cluster caching problem as minimizing the energy consumption for serving a user request by the macro base station. We show that the first sub-problem is NP-hard, and develop a polynomial-heuristic algorithm. Further, we define the mobility-oblivious version of the sub-problem, and provide a ${\textstyle {1 \over {2k }}}$ -approximation solution for the mobility-aware based on this mobility-oblivious version. In the second sub-problem, we consider the heterogeneity of small cells and define the intra-cluster caching problem. The aim of this sub-problem is the placement of content in different SCs within the cluster so that the energy consumption for the content delivery in that cluster is minimized. In particular, the second sub-problem fits the general framework of the generalized assignment problem, which allows us to exploit its rich literature, to determine the optimal solution for our problem. Through extensive simulations based on real wireless data and human mobility patterns, we demonstrate the benefits of our approach that is even up to 56% better compared to the conventional caching schemes. [ABSTRACT FROM AUTHOR]