Distributed storage systems (DESSs) are widely utilized to regulate voltages in active distribution networks with high penetration of volatile renewable energy. In this paper, the distributed multi-energy storage systems (MESSs) are integrated into the active distribution network to enhance the capability of voltage regulation by exploiting interactions among multi-energy loads. A novel distributed control strategy based on back-and-forth communication (BFC) framework is developed to optimally coordinate multiple MESSs for multi-time-step voltage regulation. To achieve the independent control of MESSs at each bus, the BFC is proposed to determine the global sensitivities of voltage violation in the whole network with respect to the power injection at each bus in only one-step iteration. These sensitivities updated in real time during BFC quantify the capability of the single MESS to mitigate voltage fluctuations across all buses, which can be used to individually inform the operation of each MESS. Distinct from traditional distributed control methods that achieve voltage regulation in a single timeslot, the proposed control strategy regulates MESSs in a multi-time-step manner to alleviate anticipated voltage violation in advance. The simulation based on a modified 11-bus radial distribution network demonstrates the effectiveness of the proposed distributed control strategy.