Context. The large-scale magnetic fields of stars and galaxies are often described in the framework of mean-field dynamo theory. At moderate magnetic Reynolds numbers, the transport coefficients defining the mean electromotive force can be determined from simulations. This applies analogously also to passive scalar transport. Aims. We investigate the mean electromotive force in the kinematic framework, that is, ignoring the back-reaction of the magnetic field on the fluid velocity, under the assumption of axisymmetric turbulence determined by the presence of either rotation, density stratification, or both. We use an analogous approach for the mean passive scalar flux. As an alternative to convection, we consider forced turbulence in an isothermal layer. When using standard ansatzes, the mean magnetic transport is then determined by nine, and the mean passive scalar transport by four coefficients. We give results for all these transport coefficients. Methods. We use the test-field method and the test-scalar method, where transport coefficients are determined by solving sets of equations with properly chosen mean magnetic fields or mean scalars. These methods are adapted to mean fields which may depend on all three space coordinates. Results. We find the anisotropy of turbulent diffusion to be moderate in spite of rapid rotation or strong density stratification. Contributions to the mean electromotive force determined by the symmetric part of the gradient tensor of the mean magnetic field, which were ignored in several earlier investigations, turn out to be important. In stratified rotating turbulence, the a effect is strongly anisotropic, suppressed along the rotation axis on large length scales, but strongly enhanced at intermediate length scales. Also the Omega x (J) over bar effect is enhanced at intermediate length scales. The turbulent passive scalar diffusivity is typically almost twice as large as the turbulent magnetic diffusivity. Both magnetic and passive scalar diffusion are slightly enhanced along the rotation axis, but decreased if there is gravity. Conclusions. The test-field and test-scalar methods provide powerful tools for analyzing transport properties of axisymmetric turbulence. Future applications are proposed ranging from anisotropic turbulence due to the presence of a uniform magnetic field to inhomogeneous turbulence where the specific entropy is nonuniform, for example. Some of the contributions to the mean electromotive force which have been ignored in several earlier investigations, in particular those given by the symmetric part of the gradient tensor of the mean magnetic field, turn out to be of significant magnitude.