  A comparison between polarization-transfer to a bound proton in quasi-free
kinematics by the A$(\vec{e},e'\vec p)$ knockout reaction and that in elastic
scattering off a free proton can provide information on the characteristics of
the bound proton. In the past the reported measurements have been compared to
those of a free proton with zero initial momentum. We introduce, for the first
time, expressions for the polarization-transfer components when the proton is
initially in motion and compare them to the $^2$H data measured at the Mainz
Microtron (MAMI). We show the ratios of the transverse ($P_x$) and longitudinal
($P_z$) components of the polarization transfer in $^2\textrm{H}(\vec{e},e'\vec
p)\textrm{n}$, to those of elastic scattering off a "moving proton", assuming
the proton's initial (Fermi) momentum equals the negative missing momentum in
the measured reaction. We found that the correction due to the proton motion is
up to 20\% at high missing momentum.
  However the effect on the double ratio
$\frac{(P_x/P_z)^A}{(P_x/P_z)^{^1\!\textrm{H}}}$ is largely canceled out, as
shown for both $^2$H and $^{12}$C data. This implies that the kinematics is not
the primary cause for the deviations between quasi-elastic and elastic
scattering reported previously.
