This work delves into the intriguing realm of fluid–structure interaction (FSI) within a semicircular grooved channel featuring the presence of elastic thin baffles. In this thermal system of cavity-shape, bottom heating is coupled with cooling at both sides of the top protruded ends. To augment the complexity of the problem, symmetrically suspended flexible elastic fins are incorporated into the channel’s top wall. The driving force behind the natural convection within the channel arises from the interplay of differential heating. Employing advanced numerical analysis techniques, specifically the Finite Element Method (FEM) and the Arbitrary Lagrangian–Eulerian (ALE) approach, the heat transfer performance is explored, by encompassing scenarios involving both rigid and flexible baffles, considering a range of Rayleigh numbers (Ra) spanning from 5 × 104 to 5 × 105, while keeping the fin’s length constant. The outcomes of this study unveil a compelling observation: oscillatory higher heat transfer rates manifest at elevated Ra values, providing valuable insights into the behavior of prolonged-running systems influenced by fluid–structure interactions.