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 (μW) frequencies. In particular, 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 μW-mmW networks is proposed. The proposed approach leverages device-level caching along with the capabilities of dual-mode base stations to minimize handover failures and provide seamless mobility. 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. Then, the impact of caching on the number of handovers (HOs) and the average handover failure (HOF) is analyzed. The derived analytical results suggest that content caching will reduce the HOF and enhance the mobility management in heterogeneous wireless networks with mmW capabilities. Numerical results corroborate the analytical derivations and show that the proposed solution provides significant reductions in the average HOF, reaching up to 45%, for mobile users moving with relatively high speeds.