  In this paper, we study a multivariate version of the generalized counting
process (GCP) and discuss its various time-changed variants. The time is
changed using random processes such as the stable subordinator, inverse stable
subordinator, and their composition, tempered stable subordinator, gamma
subordinator $etc.$ Several distributional properties that include the
probability generating function, probability mass function and their governing
differential equations are obtained for these variants. It is shown that some
of these time-changed processes are L\'evy and for such processes we have
derived the associated L\'evy measure. The explicit expressions for the
covariance and codifference of the component processes for some of these
time-changed variants are obtained. An application of the multivariate
generalized space fractional counting process to shock models is discussed.
