Understanding and predicting human cooperative behaviour and belief dynamics remains a major challenge both from the scientific and practical perspectives. Because of the complexity and multiplicity of material, social and cognitive factors involved, both empirical and theoretical work tends to focus only on some snippets of the puzzle. Recently, a mathematical theory has been proposed that integrates material, social and cognitive aspects of behaviour and beliefs dynamics to explain how people make decisions in social dilemmas within heterogeneous groups. Here we apply this theory in two countries, China and Spain, through four long-term behavioural experiments utilising the Common Pool Resources game and the Collective Risk game. Our results show that material considerations carry the smallest weight in decision-making, while personal norms tend to be the most important factor. Empirical and normative expectations have intermediate weight in decision-making. Cognitive dissonance, social projection, logic constraints and cultural background play important roles in both decision-making and beliefs dynamics. At the individual level, we observe differences in the weights that people assign to factors involved in the decision-making and belief updating process. We identify different types of prosociality and rule-following associated with cultural differences, various channels for the effects of messaging, and culturally dependent interactions between sensitivity to messaging and conformity. Our results can put policy and information design on firmer ground, highlighting the need for interventions tailored to the situation at hand and to individual characteristics. Overall, this work demonstrates the theoretical and practical power of the theory in providing a more comprehensive understanding of human behaviour and beliefs.