Single-phase concentrated solid-solution alloys (SP-CSAs), including high entropy alloys (HEAs), are compositionally complex but structurally simple, and provide a playground of tailoring material properties through modifying their compositional complexity. The recent progress in understanding the compositional effects on the energy and mass transport properties in a series of face-centered-cubic SP-CSAs is the focus of this review. Relatively low electrical and thermal conductivities, as well as small separations between the interstitial and vacancy migration barriers have been generally observed, but largely depend on the alloying constituents. We further discuss the impact of such intrinsic transport properties on their irradiation response; the linkage to the delayed damage accumulation, slow defect aggregation, and suppressed irradiation induced swelling and segregation has been presented. We emphasize that the number of alloying elements may not be a critical factor on both transport properties and the defect behaviors under ion irradiations. The recent findings have stimulated novel concepts in the design of new radiation-tolerant materials, but further studies are demanded to enable predictive models that can quantitatively bridge the transport properties to the radiation damage.