  Within the generalized hydrodynamics (GHD) formalism for quantum integrable
models, it is possible to compute simple expressions for a number of
correlation functions at the Eulerian scale. Specializing to integrable
relativistic field theories, we show the same correlators can be computed as a
sum over form factors, the GHD regime corresponding to the leading contribution
with one particle-hole pair on a finite energy-density background. The
thermodynamic bootstrap program (TBP) formalism was recently introduced as an
axiomatic approach to computing such finite-energy-density form factors for
integrable field theories. We derive a new axiom within the TBP formalism from
which we easily recover the predicted GHD Eulerian correlators. We also compute
higher form factor contributions, with more particle-hole pairs, within the
TBP, allowing for the computation of correlation functions in the diffusive,
and beyond, GHD regimes. The two particle-hole form factors agree with
expressions recently conjectured within the~GHD.
