We describe the kernel of the canonical map from the Floyd boundary of a relatively hyperbolic group to its Bowditch boundary. Using the Floyd completion we further prove that the property of relative hyperbolicity is invariant under quasi-isometric maps. If a finitely generated group $H$ admits a quasi-isometric map $\varphi$ into a relatively hyperbolic group $G$ then $H$ is itself relatively hyperbolic with respect to a system of subgroups whose image under $\varphi$ is situated within a uniformly bounded distance...