  We calculate the four-wave mixing (FWM) in a Bose-Einstein condensate system
having multiple spin wave packets that are initially overlapping in physical
space, but have nonvanishing relative momentum that cause them to recede from
one another. Three receding condensate atom wave packets can result in
production of a fourth wave packet by the process of FWM due to atom-atom
interactions. We consider cases where the four final wave packets are composed
of 1, 2, 3 and 4 different internal spin components. FWM with 1- or 2-spin
state wave packets is much stronger than 3- or 4-spin state FWM, wherein two of
the coherent moving BEC wave packets form a polarization-grating that rotates
the spin projection of the third wave into that of fourth diffracted wave (as
opposed to the 1- or 2-spin state case where a regular density-grating is
responsible for the diffraction). Calculations of FWM for $^{87}$Rb and
$^{23}$Na condensate systems are presented.
