Multimodal Emotion–Cause Triplet Extraction in Conversations (MECTEC) is fundamental for fine-grained affect understanding, yet it remains challenging in multi-turn, multi-speaker settings. Existing methods often make locally plausible predictions but struggle to maintain conversation-level consistency under within-speaker emotion shifts and core events. To address this, we propose ECFlow, a unified framework that combines appraisal-guided structured generation with graph-structured reinforcement learning. ECFlow operationalizes cognitive appraisal theory into a controllable intermediate reasoning trace and constructs UMECS, a unified supervision dataset with cognitively grounded traces. It then lifts predicted and gold triplets into an Emotion–Cause Flow Graph and optimizes verifiable, structure-aware rewards for emotion-shift coherence and core-event consistency, together with task-oriented triplet rewards. Experiments on public MECTEC benchmarks show that ECFlow consistently outperforms strong baselines, achieving state-of-the-art triplet extraction and improved structure-aware metrics on emotion shifts and core events. Our code and dataset are available at https://anonymous.4open.science/r/ECFlow-E908.