The contact formulation is strictly valid only for stationary problems. It is still possible to use contact modeling in a time-dependent analysis, as long as inertial effects are not important in the contact region or by using a sufficiently short time step. In practice, this means that you can solve both quasi-static problems and truly dynamic problems, as long as situations with impact are avoided. When in doubt, try to do an a posteriori check of conservation of momentum and energy to ensure that the solution is acceptable.