Species sampling processes provide a cornerstone for random discrete distributions and exchangeable sequences. Yet, analyzing data from distinct, though related, sources, a broader notion of probabilistic invariance is required, and partial exchangeability represents the natural choice. Over the past two decades many dependent nonparametric priors have been proposed- including hierarchical, nested and additive processes-in this setting. However, a unifying framework remains lacking. We address this by introducing multivariate species sampling processes, a general class of nonparametric priors that encompasses most existing constructions. They are characterized by their partially exchangeable partition probability function, encoding the induced multivariate clustering structure. We establish their core distributional properties and analyze their dependence structure, demonstrating that borrowing of information across groups is entirely determined by shared ties. This yields new insights into their learning mechanisms, including a principled explanation of the correlation structure induced by existing models. Besides providing a cohesive theoretical foundation, our approach serves as a constructive basis for designing new models aimed at capturing even richer dependence structures beyond the framework of multivariate species sampling processes.