In recent years, Molten Salt Reactors (MSRs), one of the Generation IV concepts, have noticeably gained in interest. Several molten salt reactor types use fuel in liquid form, bringing various advantages, notably in terms of fuel cycle. However, simulating the evolution of liquid fuel under irradiation necessitates specific tools and methods, for example to simulate the removal of insoluble fission products from the salt mixture or continuous refueling of the reactor. For this purpose, the Serpent 2-based procedure EQL0D has been developed. It is designed for both equilibrium and finite-time burn-up calculations in liquid-fueled molten salt reactors. This paper first details the most important methods implemented in the procedure to simulate liquid-fuel systems, after which the adequacy of the procedure is verified by comparing benchmark results with a comparable code. Examples of applications of this tool to the start-up and transition to closed fuel cycle of the historical single- and two-fluid Molten Salt Breeder Reactors as well as the more recent Molten Salt Fast Reactor, a fast-spectrum concept, are then presented. The obtained results show the challenge faced by thorium-cycle breeder MSRs to transition from available nuclear fuels to a closed thorium cycle, as many candidate fuels prove to be unusable for the transition to a closed cycle.