This paper generalizes a limit theorem of G. Huisken [ibid. 20, 237-266 (1984; Zbl 0556.53001)] and J. Urbas on the outward flow of hypersurfaces in \({\mathbb{R}}^{n+1}\) by mean curvature from the convex to the star shaped situation. Also, the (first) mean curvature is replaced by more general functions of the principal curvatures. The method is different and provides a fine piece of fruitful combination of differential geometry with PDE theory. Fairly intimate properties of hypersurface immersions like Codazzi tensors and their second covariant derivatives have to be applied to get the necessary uniform estimates for the parabolic flow equation.