We derive a phase field crystal model that couples the diffusive evolution of a microscopic structure with the fast dynamics of a macroscopic velocity field, explicitly accounting for the relaxation of elastic excitations. This model captures better than previous formulations the dynamics of complex interfaces and dislocations in single crystals as well as grain boundary migration in poly-crystals where the long-range elastic field is properly relaxed. The proposed model features a diffusivity that depends non-linearly on the local phase. It induces more localized interfaces between a disordered phase (liquid-like) and an ordered phase, e.g., stripes or crystal lattices. For stripes, the interface dynamics are shown to be strongly anisotropic. We also show that the model is able to evolve the classical PFC at mechanical equilibrium. However, in contrast to previous approaches, it is not restricted to a single-crystal configuration or small distortions from a fixed reference lattice. To showcase the capabilities of this approach, we consider a few examples, from the annihilation of dislocation loops in a single crystal at mechanical equilibrium to the relaxation of a microstructure including crystalline domains with different orientations and grain boundaries. During the self-annihilation of a mixed type dislocation loop (i.e., not shear or prismatic), long-range elastic effects cause the loop to move out of plane before the annihilation event.