One of the most promising approaches to overcome the uncertainty and dynamic channel variations of millimeter wave (mmW) communications is to deploy dual-mode base stations that integrate both mmW and microwave ($μ$W) frequencies. If properly designed, such dual-mode base stations can enhance mobility and handover in highly mobile wireless environments. In this paper, a novel approach for analyzing and managing mobility in joint $μ$W-mmW networks is proposed. The proposed approach leverages device-level caching along with the capabilities of dual-mode base stations to minimize handover failures, reduce inter-frequency measurement energy consumption, and provide seamless mobility in emerging dense heterogeneous networks. First, fundamental results on the caching capabilities, including caching probability and cache duration are derived for the proposed dual-mode network scenario. Second, the average achievable rate of caching is derived for mobile users. Third, the proposed cache-enabled mobility management problem is formulated as a dynamic matching game between mobile user equipments (MUEs) and small base stations (SBSs). The goal of this game is to find a distributed handover mechanism that subject to the network constraints on HOFs and limited cache sizes, allows each MUE to choose between executing an HO to a target SBS, being connected to the macrocell base station (MBS), or perform a transparent HO by using the cached content. The formulated matching game allows capturing the dynamics of the mobility management problem caused by HOFs. To solve this dynamic matching problem, a novel algorithm is proposed and its convergence to a two-sided dynamically stable HO policy is proved. Numerical results corroborate the analytical derivations and show that the proposed solution will provides significant reductions in both the HOF and energy consumption by MUEs.