As modern electronic devices become smaller and more highly integrated, stable thermal management is emerging as a key development approach, including in applications considering mechanical deformation. In this study, a flexible heat-dissipating sheet was developed using composites of insulating graphene (I-Gr), plate-like boron nitride (BN-P), and aggregated spherical BN (BN-A) based on a high-elasticity styrene-(ethylene-butylene)-styrene (SEBS) elastomer. The unique asymmetric two-dimensional layered structure of I-Gr and BN improved the heat transfer properties of the composite by maintaining the continuity of the heat-conducting network despite tensile deformation. In addition, the spherical shape and disordered structure of the aggregated BN-A promoted the formation of an extended heat-conducting path and enhanced the bonding between the fillers. At the optimal composition, the composite maintained an initial thermal conductivity (TC) of 2.0 W m^-1 K^-1 or higher, and the TC reduction (ΔTC) was less than 8% and 10% at 50% and 100% elongation, respectively, demonstrating excellent TC stability. In addition, owing to the interfacial affinity and network reinforcing effect of I-Gr, the TC performance and structural stability were maintained even after 500 cycles of 50% tensile strain and 400% elongation. In contrast, the CNT-based composite showed limitations such as low initial TC, large ΔTC, and low elongation. This study presents a design strategy for a heat-dissipating material with high elasticity, high TC, and excellent durability, offering considerable potential for use in next-generation flexible electronic devices such as wearable electronics, freeform displays, and soft robotics.