  A discrete-time quantum walk (QW) is essentially a unitary operator driving
the evolution of a single particle on the lattice. Some QWs have familiar
physics PDEs as their continuum limit. Some slight generalization of them
(allowing for prior encoding and larger neighbourhoods) even have the curved
spacetime Dirac equation, as their continuum limit. In the $(1+1)-$dimensional
massless case, this equation decouples as scalar transport equations with
tunable speeds. We characterise and construct all those QWs that lead to scalar
transport with tunable speeds. The local coin operator dictates that speed; we
provide concrete techniques to tune the speed of propagation, by making use
only of a finite number of coin operators---differently from previous models,
in which the speed of propagation depends upon a continuous parameter of the
quantum coin. The interest of such a discretization is twofold : to allow for
easier experimental implementations on the one hand, and to evaluate ways of
quantizing the metric field, on the other.
