  In three-dimensional gravity, we discuss the relation between the
Fefferman--Graham gauge, the Bondi gauge and the Eddington--Finkelstein type of
gauge, often referred to as the derivative expansion, involved in the
fluid/gravity correspondence. Starting with a negative cosmological constant,
for each gauge, we derive the solution space and the residual gauge
diffeomorphisms. We construct explicitly the diffeomorphisms that relate the
various gauges, and establish the precise matching of their boundary data. We
show that Bondi and Fefferman--Graham gauges are equivalent, while the
fluid/gravity derivative expansion, originating from a partial gauge fixing,
exhibits an extra unspecified function that encodes the boundary fluid
velocity. The Bondi gauge turns out to describe a subspace of the derivative
expansion's solution space, featuring a fluid in a specific hydrodynamic frame.
We pursue our analysis with the Ricci-flat limit of the Bondi gauge and of the
fluid/gravity derivative expansion. The relations between them persist in this
limit, which is well-defined and non-trivial. Moreover, the flat limit of the
derivative expansion maps to the ultra-relativistic limit on the boundary. This
procedure allows to unravel the holographic properties of the Bondi gauge for
vanishing cosmological constant, in terms of its boundary Carrollian dual
fluid.
