We extend recent advances in the numerical time-integration of contacting elastodynamics [Li et al. 2020] to build a new framework, called Injective Deformation Processing (IDP), for the robust solution of a wide range of mesh deformation problems requiring injectivity. IDP solves challenging 3D (and 2D) geometry processing and animation tasks on meshes, via artificial time integration, with guarantees of both non-inversion and non-overlap. To our knowledge IDP is the first framework for 3D deformation processing that can efficiently guarantee globally injective deformation without geometric locking. We demonstrate its application on a diverse set of problems and show its significant improvement over state-of-the-art for globally injective 3D deformation.