The more nuanced the declarative process constraints discovered from an event log, the more likely they are to capture meaningful business knowledge, thus fostering the application of declarative process modeling in practice. Branching the activation (source) or the target of the constraints, that is, allowing more than one event type to appear as the source or the target, is a typical way to increase their expressivity. For the discovery of Declare constraints, only the case of branching the constraint target considering the inclusive disjunction policy has been considered. In this paper, we present CBDeclare, a comprehensive approach to branched declarative process constraints, contributing in two key dimensions. First, we define a semantics of both source- and target-branched Declare constraints considering different branching policies. Second, we devise methods to discover the newly defined Declare constraint types from an event log. Our solution leverages the SIESTA framework’s scalable and incremental infrastructure for event processing, achieving significant performance gains in mining these extended constraint when compared to the solutions for target-branched constraint discovery available in the literature.