  Scalar forces "screened" by the Vainshtein mechanism may hold the key to
understanding the cosmological expansion of our universe, while predicting new
and exciting features in the interaction between massive bodies. Here we
explore the dynamics of the Vainshtein screening mechanism, focusing on the
decoupling limit of the DGP braneworld scenario and dRGT massive gravity. We
show that there is a vast set of initial conditions whose evolution is well
defined and which are driven to the static, screening solutions of these
theories. Screening solutions are stable and behave coherently under small
fluctuations: they oscillate and eventually settle to an equilibrium
configuration, the timescale for the oscillations and damping being dictated by
the Vainshtein radius of the screening solutions. At very late times, a
power-law decay ensues, in agreement with known analytical results. However, we
also conjecture that physically interesting processes such as gravitational
collapse of compact stars may not possess a well-posed initial value problem.
Finally, we construct solutions with nontrivial multipolar structure describing
the screening field of deformed, asymmetric bodies and show that higher
multipoles are screened more efficiently than the monopole component.
