In recent years, the electromagnetic rail launcher (ERL) technology has garnered widespread attention in the field of launch systems due to its outstanding performance. During ERL system operation, a large pulsed electric current flows through the system, sharply accelerating the armature to a high speed within an extremely short period, accompanied by a rapid temperature increment. This process involves complex multi-physical phenomena, posing challenges to the design and simulation of ERL systems. We propose a dynamic simulation solution for the ERL launch process through an electromagnetic-thermal-kinematics cycle. In the electric-thermal coupling simulation, the temperature-dependent electrical conductivity is considered. Joule heat produced by current is employed as the heat source for the temperature field, enhancing the accuracy of the thermal simulation. In the electromagnetic-kinematics cycle, integrating the Lorentz force acting on the armature directly simulates the force situation of the ERL propulsion. Based on the designed dynamic simulation process for the multi-physics fields of ERL systems, the accuracy of the proposed method has been validated through simulations involving square and C-type armature ERL systems, as well as laboratory measurements. Unrestricted by the limitations of control equations and solution processes, the proposed method enables flexible simulation of ERL systems.