  In a seminal work, S.A.R. Horsley and collaborators [S.A.R. Horsley {\em et
al.}, Nature Photon. {\bf 9}, 436 (2015)] have shown that, in the framework of
non-Hermitian extensions of the Schr\"odinger and Helmholtz equations, a
localized complex scattering potential with spatial distributions of the real
and imaginary parts related to one another by the spatial Kramers-Kronig
relations are reflectionless and even invisible under certain conditions. Here
we consider the scattering properties of Kramers-Kronig potentials for the
discrete version of the Schr\"odinger equation, which generally describes wave
transport on a lattice. Contrary to the continuous Schr\"odinger equation, on a
lattice a stationary Kramers-Kronig potential is reflective. However, it is
shown that a slow drift can make the potential invisible under certain
conditions.
