An improved characteristic based split (CBS) finite element method is proposed for both compressible and incompressible flows with moving boundaries. By this method, the Navier-Stokes equations in the Eulerian description are taken as the governing equations of the moving boundary problems. The segment spring analogy method is used for mesh movement to capture the moving boundaries and preserve the mesh quality during computation. Moreover, a projection procedure is proposed to exchange the flow information between different meshes, and then a new CBS solution procedure is developed for the moving mesh. Two types of moving boundary problems, including the incompressible flow induced by an in-line oscillating circular cylinder in water at rest and the inviscid transonic flow past a NACA 0012 airfoil with pitching oscillation, are used to test the proposed method. In both cases, a good agreement is obtained between the computed results and the experimental/numerical results in other literatures. Combined with suitable solvers for solid mechanics, the present CBS finite element method could be applied to the compressible and incompressible flow-structure interaction problems.