Abstract We introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal spheres ( 𝕊 n , [ g 𝕊 n ] ) $(\mathbb {S}^n,[g_{\mathbb {S}^n}])$ , it converges to the standard sphere up to a Möbius diffeomorphism. These arguments can be applied to obtain extinction profiles of solutions of some fractional porous medium equations. In the end, we use this fractional fast diffusion equation, together with its extinction profile and some estimates of its extinction time, to improve a Sobolev inequality via a quantitative estimate of the remainder term.