  We describe the dynamics of two-dimensional relativistic and Carrollian
fluids. These are mapped holographically to three-dimensional locally anti-de
Sitter and locally Minkowski spacetimes, respectively. To this end, we use
Eddington-Finkelstein coordinates, and grant general curved two-dimensional
geometries as hosts for hydrodynamics. This requires to handle the conformal
anomaly, and the expressions obtained for the reconstructed bulk metrics
incorporate non-conformal-fluid data. We also analyze the freedom of choosing
arbitrarily the hydrodynamic frame for the description of relativistic fluids,
and propose an invariant entropy current compatible with classical and extended
irreversible thermodynamics. This local freedom breaks down in the dual
gravitational picture, and fluid/gravity correspondence turns out to be
sensitive to dissipation processes: the fluid heat current is a necessary
ingredient for reconstructing all Banados asymptotically anti-de Sitter
solutions. The same feature emerges for Carrollian fluids, which enjoy a
residual frame invariance, and their Barnich-Troessaert locally Minkowski
duals. These statements are proven by computing the algebra of surface
conserved charges in the fluid-reconstructed bulk three-dimensional spacetimes.
