  The late time physics of interacting QFTs at finite temperature is controlled
by hydrodynamics. For CFTs this implies that heavy operators -- which are
generically expected to create thermal states -- can be studied
semiclassically. We show that hydrodynamics universally fixes the OPE
coefficients $C_{HH'L}$, on average, of all neutral light operators with two
non-identical heavy ones, as a function of the scaling dimension and spin of
the operators. These methods can be straightforwardly extended to CFTs with
global symmetries, and generalize recent EFT results on large charge operators
away from the case of minimal dimension at fixed charge. We also revisit
certain aspects of late time thermal correlators in QFT and other diffusive
systems.
