In wireless multi-hop networks, a fundamental problem is to disseminate continuous data traffic from a source node to all other network nodes, which is known as the broadcast problem. Such a problem becomes even more complicated in wireless multi-hop networks with energy-harvesting capabilities at nodes when facing the interaction between stochastics of traffic arrivals at the source and randomness of energy-harvesting process at nodes. In such networks, the energy consumable at a node cannot exceed the amount of the energy harvested at the node. In this paper, we investigate the throughput-optimal broadcast problem in time-varying directed acyclic wireless multi-hop networks with such energy harvesting constraints. The topologies of such networks change dynamically with time while satisfying the directed acyclic property and the energy arrival time and harvested amount at a node are random, which causes the consumable energy in each time slot to fluctuate with time. Existing throughput-optimal broadcast algorithms did not consider such energy-harvesting constraints in their designs and therefore their throughput-optimal properties do not hold anymore in such a network. In this paper, we characterize the energy-harvesting uncertainties at nodes by using time-varying per-slot-based supportable transmission rates of wireless links. We consider the time-varying property of supportable link transmission rates caused by energy-harvesting dynamics in the per-slot transmission scheduling and propose an online max-weight broadcast algorithm. We derive a tight upper bound of broadcast capacity of the wireless networks under study in this paper and further prove that our proposed algorithm is throughput-optimal. We evaluate the throughput and latency performance of the proposed algorithm by simulations and the simulation results affirm our theoretical analysis.