I consider the non-equilibrium time evolution after quantum quenches with simple inhomogeneities.
Specifically, I discuss two situations: the time evolution of a piecewise homogeneous state
under a translationally invariant Hamiltonian, and the time evolution of a homogeneous state
under a Hamiltonian with a localised defect. I present some recent results on the
corresponding late-time dynamics, and point out the similarities between the two situations.
Finally, I show that, in the presence of a defect, the late-time expectation values of the
local observables are not correctly described by a diagonal ensemble.