The fourth cumulant for the random sum of random vectors is considered. A formula is presented for the general case when the aggregating variable is independent of the random vectors. Two important special cases are considered. In the first one, multivariate skew-normal random vectors are considered that are aggregated by a Poisson variable. The second case deals with multivariatea symmetric generalized Laplace random vectors and aggregation is made by a negative binomial variable. There is a well-established relation between asymmetric Laplace motion and negative binomial process that corresponds to the invariance principle of the random sum of random vectors for the generalized asymmetric Laplace distribution. We explore this relation and provide a multivariate continuous time version of the results.