  We revisit the magneto-hydrodynamics in (2+1) dimensions and confirm that it
is consistent with the quasi-normal modes of the (3+1) dimensional dyonic black
holes in the most general set-up with finite density, magnetic field and wave
vector. We investigate all possible modes (sound, shear, diffusion, cyclotron
etc.) and their interplay. For the magneto-hydrodynamics we perform a complete
and detailed analysis correcting some prefactors in the literature, which is
important for the comparison with quasi-normal modes. For the quasi-normal mode
computations in holography we identify the independent fluctuation variables of
the dyonic black holes, which is nontrivial at finite density and magnetic
field. As an application of the quasi-normal modes of the dyonic black holes we
investigate a transport property, the diffusion constant. We find that the
diffusion constant at finite density and magnetic field saturates the lower
bound at low temperature. We show that this bound can be understood from the
pole-skipping point.
