  We comprehensively study Galilean and Carrollian hydrodynamics on arbitrary
backgrounds, in the presence of a matter/charge conserved current. For this
purpose, we follow two distinct and complementary paths. The first is based on
local invariance, be it Galilean or Carrollian diffeomorphism invariance,
possibly accompanied by Weyl invariance. The second consists in analyzing the
relativistic fluid equations at large or small speed of light, after choosing
an adapted gauge, ADM-Zermelo for the former and Papapetrou-Randers for the
latter. Unsurprisingly, the results agree, but the second approach is superior
as it effortlessly captures more elaborate situations with multiple degrees of
freedom. It furthermore allows to investigate the fate of hydrodynamic-frame
invariance in the two limits at hand, and conclude that its breaking (in the
Galilean) or its preservation (in the Carrollian) are fragile consequences of
the behaviour of transport attributes at large or small $c$. Both methods do
also agree on the doom of Noetherian currents generated in the relativistic
theory by isometries: non-trivial currents are not always guaranteed in
Newton-Cartan or Carroll spacetimes as a consequence of Galilean or Carrollian
isometries. Comparison of Galilean and Carrollian fluid equations exhibits a
striking but often superficial resemblance, which we comment in relation to
black-hole horizon dynamics, awkwardly akin to Navier-Stokes equations. This
congruity is authentic in one instance though and turns out then to describe
Aristotelian dynamics, which is the last item in our agenda.
