There are numerous applications in science and engineering that involve time-dependent deformations, e.g., fluid-structure interaction, cardiovascular modeling, metal forming, and crack propagation. For such problems, the geometric domain and the applied load change as a function of time. Hence the mesh must be updated at each time step in order to remain a valid approximation to the geometry. Mesh warping is the procedure for which the mesh is mapped from the source domain to a target domain. There are several benefits to using mesh warping instead of remeshing to generate a valid mesh. These benefits include smoothness of the mesh deformation (by preserving themesh topology); avoidance of the need to re-interpolate the PDE solution from one mesh onto another (i.e., since the mesh topology is being held fixexd); the lack of accumulation error (i.e., from not re-interpolating the PDE solution), and better computational efficiency. In this talk, we propose a high-order mesh warping algorithm for tetrahedral meshes based on a finite element formulation for hyperelastic materials. We employ the two-parameter incompressible Mooney-Rivlin model with appropriate material properties to represent the continuum model. We start with an undeformed tetrahedral mesh and the corresponding deformed surface mesh. We employ Newton iteration to solve the nonlinear elasticity equations obtained from the Mooney-Rivlin model and equilibrium conditions; the solution to the nonlinear elasticity equations then yields the deformed mesh. To obtain an initial feasible point for the Newton iteration, we solve a finite element formulation of the linear elastic model and employ a high-order iterative stiffening procedure to untangle the mesh (when needed). We safeguard the Newton step with a line search based on the scaled Jacobian. We demonstrate the convergence and stability of our finite element solver and generate several 3D dynamic high-order meshes using our algorithm. The examples stem from cardiac modeling and mechanical engineering applications.