In this work, we propose an arbitrary Lagrangian Eulerian (ALE) mesh-free method that utilizes the radial basis function-finite difference (RBF-FD) method for solving diffusion-reaction equations on evolving surfaces. The surface evolution law is determined by a forced mean curvature flow (FMCF). We develop a parametric RBF-FD method to solving the FMCF. The advantage of the proposed method is that it utilizes a suitable tangential velocity in the equation to maintain a fairly uniform distribution of nodes during the surface evolution. The tangential velocity also allows us to construct an ALE RBF-FD scheme for the surface PDEs, which performs better than the purely Lagrangian method. Furthermore, we discuss how to deal with the boundary conditions for both FMCF and PDEs on surfaces, and present a ghost node approach to efficiently discretize the boundary conditions. Numerical experiments are shown, which not only demonstrate the effectiveness of the tangential velocity, but also illustrate some applications.