  Collisionless relativistic shocks have been the focus of intense theoretical
and numerical investigations in recent years. The acceleration of particles,
the generation of electromagnetic microturbulence and the building up of a
shock front are three interrelated essential ingredients of a relativistic
collisionless shock wave. In this paper we investigate two issues of importance
in this context: (1) the transport of suprathermal particles in the excited
microturbulence upstream of the shock and its consequences regarding particle
acceleration; (2) the preheating of incoming background electrons as they cross
the shock precursor and experience relativistic oscillations in the
microturbulent electric fields. We place emphasis on the importance of the
motion of the electromagnetic disturbances relatively to the background plasma
and to the shock front. This investigation is carried out for the two major
instabilities involved in the precursor of relativistic shocks, the
filamentation instability and the oblique two stream instability. Finally, we
use our results to discuss the maximal acceleration at the external shock of a
gamma-ray burst; we find in particular a maximal synchrotron photon energy of
the order of a few GeV.
