  We consider the non-equilibrium time evolution of a translationally invariant
state under a Hamiltonian with a localized defect. We discern the situations
where a light-cone spreads out from the defect and separates the system into
regions with macroscopically different properties. We identify the light-cone
and propose a procedure to obtain a (quasi-)stationary state describing the
late time dynamics of local observables. As an explicit example, we study the
time evolution generated by the Hamiltonian of the transverse-field Ising chain
with a local defect that cuts the interaction between two sites (a quench of
the boundary conditions alongside a global quench). We solve the dynamics
exactly and show that the late time properties can be obtained with the general
method proposed.
