Soft materials with high thermal conductivity are critical for flexible electronics, energy storage and transfer, and human-interface devices and robotics. However, fundamental heat transport limitations in soft and deformable materials present significant challenges for achieving high thermal conductivity. Here, a systematic study of soft composites with solid, liquid, and solid-liquid multiphase metal fillers dispersed in elastomers reveals key strategies to tune the thermal-mechanical response of soft materials. Experiments supported by thermodynamic and kinetic modeling demonstrate that multiphase systems quickly form intermetallics that solidify and degrade mechanical response with modest gains in thermal conductivity. In contrast, liquid metal 1 This is the author manuscript accepted for publication and has undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process, which may lead to differences between this version and the Version of Record. Please cite this article as