Combinations of gyroscopic forces and nonequilibrium activity have been explored recently in rectifying energy in networks with complex geometries and topologies [Phys. Rev. X 10, 021036 (2020)2160-330810.1103/PhysRevX.10.021036]. Based on this previous work, here we study the effect of added time-periodic modulations. Numerical calculations show that the time-modulated network generates net energy transport between sites and the surroundings, even in the absence of any temperature gradients. Combining path integral formulation and diagrammatic expansion, we explain how such anomalous energy transport emerges, and show how the transport pattern in complex networks can be connected to relatively simple local structures.