We consider the problem of belief propagation in a network of communicating agents, modeled in the recently introduced Belief Revision Game (BRG) framework. In this setting, each agent expresses her belief through a propositional formula and revises her own belief at each step by considering the beliefs of her acquaintances, using belief change tools. In this paper, we investigate the extent to which BRGs satisfy some monotonicity properties, i.e., whether promoting some desired piece of belief to a given set of agents is actually always useful for making it accepted by all of them. We formally capture such a concept of promotion by a new family of belief change operators. We show that some basic monotonicity properties are not satisfied by BRGs in general, even when the agents merging-based revision policies are fully rational (in the AGM sense). We also identify some classes where they hold.