  We discuss the non-linear extension to the integrated Sachs-Wolfe effect
(ISW) resulting from the divergence of the large scale structure momentum
density field. The non-linear ISW effect leads to an increase in the total ISW
contribution by roughly two orders of magnitude at l ~ 1000. This increase,
however, is still below the cosmic variance limit of the primary anisotropies;
at further small angular scales, secondary effects such as gravitational
lensing and the kinetic Sunyaev-Zel'dovich (SZ) effect dominates the non-linear
ISW power spectrum. We show this second-order non-linear ISW contribution is
effectively same as the contribution previously described as a lensing effect
due to the transverse motion of gravitational lenses and well known as the
Kaiser-Stebbins effect under the context of cosmic strings. Due to geometrical
considerations, there is no significant three point correlation function, or a
bispectrum, between the linear ISW effects and its non-linear extension. The
non-linear ISW contribution can be potentially used as a probe of the
transverse velocity of dark matter halos such as galaxy clusters. Due to the
small contribution to temperature fluctuations, of order few tenths of micro
Kelvin, however, extrating useful measurements on velocities will be
challenging.
