Let \(M\) be a differentiable manifold and denote by \(\nabla\) and \(\tilde{\nabla}\) two linear connections on \(M\). \(\nabla\) and \(\tilde{\nabla}\) are said to be geodesically equivalent if and only if they have the same geodesics. A Riemannian manifold \((M,g)\) is a naturally reductive homogeneous manifold if and only if \(\nabla\) and \(\tilde{\nabla}=\nabla-T\) are geodesically equivalent, where \(T\) is a homogeneous structure on \((M,g)\) ([Tricerri F., Vanhecke L., Homogeneous Structure on Riemannian Manifolds. London Math. Soc. Lecture Note Series, vol. 83, Cambridge Univ. Press 1983]). In the present paper we prove that if it is possible to map geodesically a homogeneous Riemannian manifold \((M,g)\) onto \((M,\tilde{\nabla})\), then the map is affine. If a naturally reductive manifold \((M,g)\) admits a nontrivial geodesic mapping onto a Riemannian manifold \((\overline{M},\overline{g})\) then both manifolds are of constant cutvature. We also give some results for almost geodesic mappings \((M,g) \to (M,\tilde{\nabla})\).