To cope with the uncertainty brought by the large-scale integration of renewable energy under the goal of carbon neutrality, it is necessary to tap and utilize flexible and adjustable resources from both the source and the load side at the same time. Hence, a flexible low-carbon optimal scheduling method for distribution networks is proposed in this paper, which takes into account the participation of heat storage industrial loads in demand response. Firstly, the model of the gas turbine equipped with a flexible carbon capture device is established, and the non-convex constraint introduced by the adjustable flue gas diversion ratio is convexified. Then the model of the fused magnesium load, a representative of heat storage industrial loads, is established for its participation in demand response. The segment linearization and convexification methods are performed on the conditional productivity constraints of the fused magnesium load. On this basis, a mixed-integer linear programming model for flexible and low-carbon optimal dispatch of the distribution network is developed by using the stochastic optimization theory and solved by commercial solvers. The proposed method is verified to be able to ensure the economic operation of the distribution network while reducing carbon emissions and promoting renewable energy consumption.