Radial conformal motions are considered in conformally flat space–times and their properties are used to obtain conformal factors. The geodesic case leads directly to the conformal factor of Robertson–Walker universes. General cases admitting homogeneous expansion or orthogonal hypersurfaces of constant curvature are analyzed separately. When the two conditions above are considered together a subfamily of the Stephani perfect fluid solutions, with acceleration Fermi–Walker propagated along the flow of the fluid, follows. The corresponding conformal factors are calculated and contrasted with those associated with Robertson–Walker space–times.