Stiff or semiflexible filaments can be cross-linked to form a network structure with unusual mechanical properties, if the cross-links at network junctions have the ability to dynamically break and re-form. The characteristic rheology, arising from the competition of plasticity from the transient cross-links and nonlinear elasticity from the filament network, has been widely tested in experiments. Though the responses of a transient filament network under small deformations are relatively well understood by analyzing its linear viscoelasticity, a continuum theory adaptable for finite or large deformations is still absent. Here we develop a model for transient filament networks under arbitrary deformations, which is based on the cross-link dynamics and the macroscopic system tracking the continuously reshaping reference state. We apply the theory to explain the stress relaxation, the shape recovery after instant deformation, and the necking instability under a ramp deformation. We also examine the role of polydispersity in the mesh size of the network, which leads to a stretched exponential stress relaxation and a diffuse elastic-plastic transition under a ramp deformation. Although dynamic cross-links are taken as the source of the transient network response, the model can be easily adjusted to incorporating other factors inducing fluidization, such as filament breakage and active motion of motor cross-links, opening a way to address cell and tissue activity at the microscopic level.