  This work is a follow-up on the work of the second author with P.
Daskalopoulos and J.L. V\'{a}zquez. In this latter work, we introduced the
Yamabe flow associated to the so-called fractional curvature and prove some
existence result of mild (semi-group) solutions. In the present work, we
continue this study by proving that for some class of data one can prove
actually convergence of the flow in a more general context. We build on the
approach simplified in the book of M. Struwe.
