You seem to be looking for mapply
. Here's an example:
listA <- list(matrix(rnorm(2000), nrow=10),
matrix(rnorm(2000), nrow=10))
listB <- list(matrix(rnorm(2000), nrow=10),
matrix(rnorm(2000), nrow=10))
mapply(function(X,Y) {
sapply(1:10, function(row) cor(X[row,], Y[row,]))
}, X=listA, Y=listB)