Recently, several logics modelling evidence have been proposed in the literature. These logics often also feature beliefs. We call the process or function that maps evidence to beliefs consolidation. In this paper, we use a four-valued modal logic of evidence as a basis. In the models for this logic, agents are represented by nodes, peer connections by edges and the private evidence that each agent has by a four-valued valuation. From this basis, we propose methods of consolidating the beliefs of the agents, taking into account both their private evidence as well as their peers' opinions. To this end, beliefs are computed iteratively. The final consolidated beliefs are the ones in the point of stabilization of the model. However, it turns out that some consolidation policies will not stabilize for certain models. Finding the conditions for stabilization is one of the main problems studied here, along with other properties of such consolidations. Our main contributions are twofold: we offer a new dynamic perspective on the process of forming evidence-based beliefs, in the context of evidence logics, and we set up and address some mathematically challenging problems, which are related to graph theory and practical subject areas such as belief/opinion diffusion and contagion in multi-agent networks.