One of the classical questions about the evolution of a one
dimensional quantum system is the asymptotic rate of propagation of
the wave packet. It is usually captured through the notion of
transport exponents. Several methods were developed to estimate these
quantities in various models. However many authors only treated the
case of a state initially localized at a single site (in the discrete
setting). We show that some of these results can be extended to a
broad class of initial states with compact or even infinite support,
and explain what are the methods and obstacles to further
generalizations.