  We study asymptotically slowly varying perturbations of the AdS black brane
in Einstein's gravity with a negative cosmological constant. We allow both the
induced metric and the Brown-York stress tensor at a given radial cut-off slice
to fluctuate. These fluctuations, which determine the radial evolution of the
metric, are parametrized in terms of boundary data. We observe that the
renormalized energy-momentum tensor at any radial slice takes the standard
hydrodynamic form which is relativistically covariant with respect to the
induced metric. The RG flow of the fluid takes the form of field redefinitions
of the boundary hydrodynamic variables. To show this, up to first order in the
derivative expansion, we only need to investigate the radial flow of the
boundary data and do not need to impose constraints on them. Imposing the
constraints gives unforced nonlinear hydrodynamic equations at any radial
slice. Along the way we make a careful study of the choice of counter-terms and
hypersurfaces involved in defining the holographic RG flow, while at the same
time we do not explicitly set any boundary condition either at the cut-off or
at the horizon. We find that \eta/s does not change along the RG flow, equaling
1/(4\pi) when the future horizon is regular. We also analyze the flow of the
speed of sound and find that it diverges at the horizon.
