  Combined action of helical motions of plasma (the $\alpha$ effect) and
non-uniform (differential) rotation is a key dynamo mechanism of solar and
galactic large-scale magnetic fields. Dynamics of magnetic helicity of
small-scale fields is a crucial mechanism in a nonlinear dynamo saturation
where turbulent magnetic helicity fluxes allow to avoid catastrophic quenching
of the $\alpha$ effect. The convective zone of the Sun and solar-like stars as
well as galactic discs are the source for production of turbulent magnetic
helicity fluxes. In the framework of the mean-field approach and the spectral
$\tau$ approximation, we derive turbulent magnetic helicity fluxes using the
Coulomb gauge in a density-stratified turbulence. The turbulent magnetic
helicity fluxes include non-gradient and gradient contributions. The
non-gradient magnetic helicity flux is proportional to a nonlinear effective
velocity (which vanishes in the absence of the density stratification)
multiplied by small-scale magnetic helicity, while the gradient contributions
describe turbulent magnetic diffusion of the small-scale magnetic helicity. In
addition, the turbulent magnetic helicity fluxes contain source terms
proportional to the kinetic $\alpha$ effect or its gradients, and also
contributions caused by the large-scale shear (solar differential rotation). We
have demonstrated that the turbulent magnetic helicity fluxes due to the
kinetic $\alpha$ effect and its radial derivative in combination with the
nonlinear magnetic diffusion of the small-scale magnetic helicity are dominant
in the solar convective zone.
