  We study potentially observable consequences of spatiotemporal discreteness
for the motion of massive and massless particles. First we describe some simple
intrinsic models for the motion of a massive point particle in a fixed causal
set background. At large scales, the microscopic swerves induced by the
underlying atomicity manifest themselves as a Lorentz invariant diffusion in
energy-momentum governed by a single phenomenological parameter, and we derive
in full the corresponding diffusion equation. Inspired by the simplicity of the
result, we then derive the most general Lorentz invariant diffusion equation
for a massless particle, which turns out to contain two phenomenological
parameters describing, respectively, diffusion and drift in the particle's
energy. The particles do not leave the light cone however: their worldlines
continue to be null geodesics. Finally, we deduce bounds on the drift and
diffusion constants for photons from the blackbody nature of the spectrum of
the cosmic microwave background radiation.
