Random measures provide flexible parameters for Bayesian nonparametric models. Given two different priors for a random measure, we develop a natural framework to investigate the rate at which the corresponding posteriors merge, as the sample size increases. We define a new distance between the laws of random measures that is built as a Wasserstein distance on the ground space of unbalanced measures, endowed with the bounded Lipschitz metric. We develop tight analytical bounds for its specification to completely random measures, including the special case of Poisson and gamma random measures. The bounds are interpreted in terms of an adapted extended Wasserstein distance between the Lévy measures and are used to investigate the merging between the posteriors of normalized gamma and generalized gamma priors. After a careful study on the identifiability of the law of the random measure, interesting asymptotic and finite-sample insights are derived without putting any assumption on the true data generating process.