The paper introduces the gas-kinetic scheme for three-dimensional (3D) flow simulation. First, under a unified coordinate transformation, the 3D gas-kinetic BGK equation is transformed into a computational space with arbitrary mesh moving velocity. Second, based on the Chapman-Enskog expansion of the kinetic equation, a local solution of gas distribution function is constructed and used in a finite volume scheme. As a result, a Navier-Stokes flow solver is developed for the low speed flow computation with dynamical mesh movement. Several test cases are used to validate the 3D gas-kinetic method. The first example is a 3D cavity flow with up-moving boundary at Reynolds number $3200$, where the periodic solutions are compared with the experimental measurements. Then, the flow evolution inside a rotating 3D cavity is simulated with the moving mesh method, where the solution differences between 2D and 3D simulation are explicitly presented. Finally, the scheme is applied to the falling plate study, where the unsteady plate tumbling motion inside water tank has been studied and compared with the experimental measurements.