  We perform explicit time-dependent classical and quantum propagation of a
spatially indirect exciton (SIX) driven by surface acoustic waves (SAWs) in a
semiconductor heterostructure device. We model the SIX dynamics at different
levels of description, from the Euler-Lagrange propagation of structureless
classical particles to unitary Schr\"odinger propagation of an electron-hole
wave packet in a mean field and to the full quantum propagation of the
two-particle complex. A recently proposed beyond mean-field self-energy
approach, adding internal virtual transitions to the c.m. dynamics, has been
generalized to time-dependent potentials and turns out to describe very well
full quantum calculations, while being orders of magnitude numerically less
demanding. We show that SAW-driven SIXs are a sensitive probe of scattering
potentials in the devices originating, for example, from single impurities or
metallic gates, due to competing length and energy scales between the SAW
elastic potential, the scattering potential, and the internal electron-hole
dynamic of the SIX. Comparison between different approximations allow us to
show that internal correlation of the electron-hole pair is crucial in
scattering from shallow impurities, where tunneling plays a major role. On the
other hand, scattering from broad potentials, i.e., with length scales
exceeding the SIX Bohr radius, is well described as the classical dynamics of a
pointlike SIX. Recent experiments are discussed in light of our calculations.
