  We explore the science prospects of a 14,000 deg$^2$ Kinematic Lensing (KL)
survey with the Dark Energy Spectroscopic Instrument (DESI) and overlapping
imaging surveys. KL infers the cosmic shear signal by jointly forward modeling
the observed photometric image and velocity field of a disk galaxy. The latter
can be constrained by placing multiple DESI fibers along the galaxy's major and
minor axis, a concept similar to the DESI Peculiar Velocity Survey. We study
multiple subset galaxy samples of the DESI Legacy Survey Data Release 9 catalog
and quantify the residual shape noise, $\sigma_\epsilon$, of each sample as a
function of cuts in $r$-band magnitude using mock observations. We conduct
simulated likelihood analyses for these galaxy samples and find that a DESI-KL
program can place highly interesting constraints on structure formation at very
low redshifts, i.e. $\sigma_8(z<0.15)$. We conclude that if the $S_8$ tension
consolidates as a phenomenon, a DESI-KL survey can provide unique insights into
this phenomenon in the very late-time Universe. Given the different footprints
of DESI and Rubin Observatory's Legacy Survey of Space and Time (LSST), lensing
results from both surveys are highly complementary and can be combined into a
joint lensing survey. We further note that DESI-KL benefits multiple additional
science cases, e.g. studies of modified gravity models when combined with
peculiar velocity surveys, and dark matter studies that are based on
galaxy-galaxy lensing of dwarf galaxies.
