  Studies of in-vacuo dispersion are the most active area of quantum-gravity
phenomenology. The way in which in-vacuo dispersion produces redshift-dependent
corrections to the time of flight of astrophysics particles depends on the
model-dependent interplay between Planck-scale effects and spacetime
curvature/expansion, and we here derive the most general formula for the
leading order redshift-dependent correction to the time of flight for the
scenario in which relativistic symmetries are deformed at the Planck scale
(DSR) for the constant-curvature case. We find that, contrary to the broken
symmetries scenario (LIV), where in principle any arbitrary form of redshift
dependence could be allowed, for the DSR scenario only linear combinations of
three possible forms of redshift dependence are allowed. We also derive a
generalization of our results to the FRW case, and discuss some specific
combinations of the three forms of redshift dependence whose investigation
might deserve priority from the quantum-gravity perspective.
