  In recent years, three exceptional discretizations of the phi^4 theory have
been discovered [J.M. Speight and R.S. Ward, Nonlinearity 7, 475 (1994); C.M.
Bender and A. Tovbis, J. Math. Phys. 38, 3700 (1997); P.G. Kevrekidis, Physica
D 183, 68 (2003)] which support translationally invariant kinks, i.e. families
of stationary kinks centred at arbitrary points between the lattice sites. It
has been suggested that the translationally invariant stationary kinks may
persist as 'sliding kinks', i.e. discrete kinks travelling at nonzero
velocities without experiencing any radiation damping. The purpose of this
study is to check whether this is indeed the case. By computing the Stokes
constants in beyond-all-order asymptotic expansions, we prove that the three
exceptional discretizations do not support sliding kinks for most values of the
velocity - just like the standard, one-site, discretization. There are,
however, isolated values of velocity for which radiationless kink propagation
becomes possible. There is one such value for the discretization of Speight and
Ward and three 'sliding velocities' for the model of Kevrekedis.
