  Using the theory of generalized hydrodynamics (GHD), we derive exact
Euler-scale dynamical two-point correlation functions of conserved densities
and currents in inhomogeneous, non-stationary states of many-body integrable
systems with weak space-time variations. This extends previous works to
inhomogeneous and non-stationary situations. Using GHD projection operators, we
further derive formulae for Euler-scale two-point functions of arbitrary local
fields, purely from the data of their homogeneous one-point functions. These
are new also in homogeneous generalized Gibbs ensembles. The technique is based
on combining a fluctuation-dissipation principle along with the exact solution
by characteristics of GHD, and gives a recursive procedure able to generate
$n$-point correlation functions. Owing to the universality of GHD, the results
are expected to apply to quantum and classical integrable field theory such as
the sinh-Gordon model and the Lieb-Liniger model, spin chains such as the XXZ
and Hubbard models, and solvable classical gases such as the hard rod gas and
soliton gases. In particular, we find Leclair-Mussardo-type infinite
form-factor series in integrable quantum field theory, and exact Euler-scale
two-point functions of exponential fields in the sinh-Gordon model and of
powers of the density field in the Lieb-Liniger model. We also analyze
correlations in the partitioning protocol, extract large-time asymptotics, and,
in free models, derive all Euler-scale $n$-point functions.
