A high-order arbitrary Lagrangian-Eulerian (ALE) oscillation-free discontinuous Galerkin scheme for one-dimensional compressible multi-material flows is proposed. It couples a conservative equation in the volume-fraction model and the Euler equations of the fluid mixture dynamics. The mesh velocity is obtained by an adaptive mesh method, which automatically concentrates mesh nodes near the regions with large gradients. This greatly reduce numerical dissipation at material interfaces. Besides, the resolution of the solution at material interfaces can be also improved. To control the oscillations, we add damping terms into the weak formulation of the system. Neither parameter in new terms has to be artificially adjusted, and the difficulties with discontinuous solutions and complexities of designing limiters can be avoided. The scheme can be efficiently applied to compressible multi-material flows with essentially non-oscillatory property and its steps are more concise than in indirect ALE methods. Although here we only consider a third-order scheme, the method can be extended to any other order by choosing suitable basis functions. Examples demonstrate the third-order accuracy and good performance of the scheme for one-dimensional compressible multi-material flows.