Caching on the edge has been recognized as an effective solution to tackle the backhaul constraint of network densification. However, most related works ignored user mobility in wireless networks, which is unreasonable under the background of network densification. For a more flexible and context-aware caching decision, the concept of caching on the edge can be extended to mobile edge computing (MEC) that enables computation and storage resources at mobile edge networks. With MEC servers deployed on base stations, a huge amount of collected radio access network context data can be analyzed and utilized to render a caching scheme adaptive to user’s context-aware information. In this regard, a novel mobility-aware coded probabilistic caching scheme is proposed for MEC-enabled small cell networks (SCNs). Different from previous mobility-aware caching schemes, user mobility and distributed storage are incorporated into a conventional probabilistic caching scheme, with the aim of throughput maximization. Based on stochastic geometry theory and a modified mobility model of discrete random jumps, the explicit expression of throughput is derived. Due to the complexity of the expression, two light-weight heuristic algorithms are provided to numerically obtain the optimal solutions. Moreover, a significant trade-off among the gains of mobility diversity, content diversity, and channel selection diversity is discussed, and we further numerically analyze how such a trade-off is influenced by user mobility, content popularity, and backhaul capacity, with some fundamental insights into the application of the proposed scheme in MEC-enabled SCNs. The superiority of our proposed scheme is demonstrated by the comparisons with the classical M most popular caching scheme and the conventional probabilistic caching scheme. Numerical results show that our proposed caching scheme achieves higher throughput than those of the other two, especially when users of intense mobility request contents, of which the popularity profile is not skewed, in MEC-enabled SCNs with poor backhaul capacity, indicating that the proposed caching scheme is a promising solution for network densification.