  Doubly Special Relativity (DSR) theories consider (quantum-gravity motivated)
deformations of the symmetries of special relativity compatible with a
relativity principle. The existence of time delays for massless particles, one
of their proposed phenomenological consequences, is a delicate question, since,
contrary to what happens with Lorentz Invariance Violation (LIV) scenarios,
they are not simply determined by the modification in the particle dispersion
relation. While some studies of DSR assert the existence of photon time delays,
in this paper we generalize a recently proposed model for time delay studies in
DSR and show that the existence of photon time delays does not necessarily
follow from a DSR scenario, determining in which cases this is so. Moreover, we
clarify long-standing questions about the arbitrariness in the choice of the
energy-momentum labels and the independence of the time delay on this choice,
as well as on the consistency of its calculation with the relative locality
paradigm of DSR theories. Finally, we show that the result for time delays is
reproduced in models that consider propagation in a noncommutative spacetime.
