This article proposes a distributed multi-period multi-energy operational model for the multi-carrier energy system. In this model, energy hubs function as distributed decision-makers and feature the synergistic interactions of generation, delivery, and consumption of coupled electrical, heating, and natural gas energy networks. The multi-period multi-energy scheduling is a challenging optimization problem due to its strong couplings and inherent nonconvexities within the multi-energy networks. The original problem is thus reformulated as a mixed integer second-order cone programming (MISOCP) and subsequently solved with a sequential second-order cone programming (SOCP) approach to guarantee a satisfactory convergence performance. Furthermore, a fully-distributed consensus-based alternating direction method of multipliers (ADMM) approach with only neighboring information exchange required is developed to optimize the multi-energy flows while considering the local energy-autonomy of heterogeneous energy hubs. The proposed methodology is performed and benchmarked on a four-hub urban multi-energy system over a 24 hourly scheduling periods. Simulation results demonstrated the superiority of the proposed scheme in system operational economy and renewable energy utilization, and also verify the effectiveness of the proposed distributed approach.