There exist several endeavors proposing a new family of extended distributions using the beta-generating technique. This is a well-known mechanism in developing flexible distributions, by embedding the cumulative distribution function (cdf) of a baseline distribution within the beta distribution that acts as a generator. Univariate beta-generated distributions offer many fruitful and tractable properties and have applications in hydrology, biology and environmental sciences amongst other fields. In the univariate cases, this extension works well, however, for multivariate cases, the beta distribution generator delivers complex expressions. In this document, the proposed extension from the univariate to the multivariate domain addresses the need of flexible multivariate distributions that can model a wide range of multivariate data. This new family of multivariate distributions, whose marginals are beta-generated distributed, is constructed with the function H(x_{1},...,x_{p})=F(G_{1}(x_{1}),G_{2}(x_{2}),...,G_{p}(x_{p})), where $G_{i}(x_{i})$ are the cdfs of the gamma (baseline) distribution and F(.) as the cdf of the Dirichlet distribution. Hence as the main example, a general model having the support [0,1]^{p} (for p variates), using the Dirichlet as the generator, is developed together with some distributional properties, such as the moment generating function. The proposed Dirichlet-generated distributions can be applied to compositional data. The parameters of the model are estimated by using the maximum likelihood method. The effectiveness and prominence of the proposed family are illustrated by analyzing simulated as well as two real datasets. A new model testing technique is introduced to evaluate the performance of the multivariate models.