  High-energy evolution equations, such as the BFKL, BK or JIMWLK equations,
aim at resumming the high-energy (next-to-)leading logarithms appearing in QCD
perturbative series. However, the standard derivations of those equations are
performed in a strict high-energy limit, whereas such equations are then
applied to scattering processes at large but finite energies. For that reason,
there is typically a slight mismatch between the leading logs resummed by those
evolution equations without finite-energy corrections and the leading logs
actually present in the perturbative expansion of any observable. That mismatch
is one of the sources of large corrections at NLO and NLL accuracy. In the case
of the BFKL equation in momentum space, that problem is solved by including a
kinematical constraint in the kernel, which is the most important finite-energy
correction. In this paper, such an improvement of kinematics is performed in
mixed-space (transverse positions and $k^+$) and with a factorization scheme in
the light-cone momentum $k^+$ (in a frame in which the projectile is
right-moving and the target left-moving). This is the usual choice of variables
and factorization scheme for the the BK equation. A kinematically improved
version of the BK equation is provided, consistent at finite energies. The
results presented here are also a necessary step towards having the high energy
limit of QCD (including gluon saturation) quantitatively under control beyond
strict leading logarithmic accuracy.
