Recent advances in the Optimal Transport theory allow to rewrite several known problems in a neat way, while providing a more general perspective. When dealing with mixture densities, or in general with intensities, such a framework naturally induces composite dissimilarities, together with corresponding Greedy Reduction and Refinement algorithms. In applications like target tracking in clutter, it is common to deal with the Mixture Reduction problem, since the optimal Bayesian recursion leads to a combinatorial explosion of hypotheses for the posterior distribution. Moreover, in the extended target case, more complex distributions are being considered to describe the features of an object, for instance the Gamma Gaussian inverse-Wishart density, which makes the reduction problem intrinsically more difficult. For the reasons above, having theoretically sound reduction algorithms results to be important for many practical problems. In this work, we will provide an optimal transport perspective to the Gamma Gaussian inverse-Wishart mixture reduction problem, together with algorithms which are suitable for real-time applications.