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.