Fluid-structure interaction simulations where solid bodies undergo large deformations re-quire special handling of the mesh motion for Arbitrarily Lagrangian-Eulerian (ALE) formu-lations. Such formulations are necessary when body-fitted meshes with certain characteristics, such as boundary layer resolution, are required to properly resolve the problem. We present an overset mesh method to accommodate such problems in which flexible bodies undergo large de-formations, or where rigid translation modes of motion occur. To accommodate these motions of the bodies through the computational domain, an overset mesh enabled ALE formulation for fluid flow is discretized with the hybridizable discontinuous Galerkin (HDG) finite element method. The overset mesh framework applied to the HDG method enables the deforming and translating dynamic meshes to maintain quality without remeshing. Verification is performed to demonstrate that optimal order convergence O (k + 1) is obtained for arbitrary overlap and approximation order k.