With the widespread integration of distributed energy resources and novel loads, the DC attributes of distribution networks are becoming increasingly pronounced. Multi-terminal flexible DC distribution networks have emerged as a trend for future distribution grids due to lower line losses, better power quality, etc. However, owing to their low damping and inertia, the multi-terminal flexible DC distribution network is vulnerable to DC faults. Analyzing the fault characteristics and calculating the fault current level is of great significance for the design of relay protection systems and the optimization of associated parameters. Throughout the fault process, the discharge paths of multiple converters are mutually coupled, and the fault characteristics are complex, which poses a great challenge to short-circuit calculations. This paper proposes a method for calculating the characteristic quantities of the whole network throughout the asymmetric short-circuit fault in a multi-terminal flexible DC distribution network. During the capacitor discharge stage, an equivalent model of the fault port is established before the control response. During the fault ride-through stage, a transfer matrix that takes into account the electrical constraints on both the AC and DC sides of the converters is proposed by combining the equivalent circuit of fully controlled converters. Finally, a simulation model of a six-terminal flexible DC distribution network is developed in PSCAD/EMTDC, and the simulation results demonstrate that the proposed method expands the calculation range from faulty branch to network-wide characteristic quantities throughout the process of asymmetric short-circuit faults, with the maximum relative error remaining below 5%.