This research investigates the intriguing phenomenon of fluid–structure interaction (FSI) within a semicircular grooved channel featuring twin flexible baffles, along with two side inlets and one central top outlet. This thermal system involves heating at the cavity’s bottom while cooling water enters through the top channel from both ends. Symmetrically suspended flexible or elastic fins from the channel’s top wall further enrich the complexity of the problem. Mixed convection within the channel is driven by the interplay of differential heating and cooling water inflow. This numerical analysis employs the Arbitrary Lagrangian–Eulerian (ALE) approach in the framework of Finite Element Method (FEM) and the study is conducted to comprehensively explore the heat transfer performance. The study considers both rigid and flexible baffles for Reynolds numbers (Re) of 100 and 200 and Richardson number (Ri) of 0.1 to 10 while maintaining length and Cauchy number (Ca) as fixed parameters. The results reveal a compelling observation: oscillatory higher heat transfer rates are observed at the higher studied values of Re and Ri, offering valuable insights into the behavior of prolonged-running systems with fluid–structure interactions.