We study a framework for robust mechanism design with multiple agents that accommodates various degrees of robustness with respect to agents' beliefs, and encompasses both the belief-free and Bayesian robustness criteria. For general \emph{belief restrictions}, we characterize the set of incentive compatible direct transfer mechanisms in general environments with interdependent values. Based on a \emph{first-order approach}, we obtain a design principle to attain incentive compatibility via `belief-based' terms. In environments that satisfy a property of \emph{generalized independence}, our results imply a \emph{robust} version of \emph{revenue equivalence}. Extending the notion of correlated information, we introduce a notion of \emph{comovement} between types and beliefs, defined based on a moment condition. Under comovement, we characterize the full set of `belief-based' terms. Based on this, we show that from Bayesian settings the following result extends to this fairly mild restriction on beliefs: any allocation rule can be implemented, even in environments without single-crossing or monotonicity. However, full rent extraction need not follow. Information rents typically remain, and they decrease monotonically as the robustness requirement is weakened.