Millimeter wave (mmWave) frequencies with rich bandwidth can greatly alleviate the surging traffic demand of wireless communication from mobile video streaming. However, the harsh propagation environment at mmWave bands and user mobility make it hard to provide a reliable service. Multibeam transmissions can be taken as an essential approach for maintaining a robust mmWave link. Meanwhile, the introduction of coded caching can further reduce the end-to-end latency and backhaul traffic. Considering the interaction between caching and transmission in mobility scenario, we develop a joint design for mmWave heterogeneous networks (HetNets), by combining multibeam transmission and maximum distance separable (MDS)-coded caching. First, a joint optimization problem of cache placement and beam pair selection is formulated, in order to maximize the backhaul traffic offload in mmWave HetNets. To solve the mixed-timescale mixed-integer nonlinear programming (MINLP) problem, we then decompose the joint problem into the long-term mobility-based proactive content caching (CC) and short-term content delivery (CD) subproblems. Subsequently, two many-to-many matching-based algorithms are proposed for tackling the CD subproblem and the convex technique is applied to deal with the CC subproblem. Simulation results demonstrate the efficacy of the proposed algorithms over benchmark schemes.