It is shown that moving averages sequences
are good in the mean for multiparameter strongly superadditive
processes
in $L_1$, and good in the p-mean for multiparameter admissible
superadditive processes in $L_p$, $1\leq p<\infty$. Also,
using a decomposition theorem in $L_p$-spaces, a.e.~convergence of
the moving averages of multiparameter superadditive
processes with respect to positive $L_{p}$-contractions, $1<p<\infty$,
is obtained.