  We derive a Moyal dynamical equation that describes exact time evolution in
generic (inhomogeneous) noninteracting spin-chain models. Assuming
quasistationarity, we develop a hydrodynamic theory. The question at hand is
whether some large-time corrections are captured by higher-order hydrodynamics.
We consider in particular the dynamics after that two chains, prepared in
different conditions, are joined together. In these situations a light cone,
separating regions with macroscopically different properties, emerges from the
junction. In free fermionic systems some observables close to the light cone
follow a universal behavior, known as Tracy-Widom scaling. Universality means
weak dependence on the system's details, so this is the perfect setting where
hydrodynamics could emerge. For the transverse-field Ising chain and the XX
model, we show that hydrodynamics captures the scaling behavior close to the
light cone. On the other hand, our numerical analysis suggests that
hydrodynamics fails in more general models, whenever a condition is not
satisfied.
