  We study the long-time behavior of the Schr{\"o}dinger flow in a
heterogeneous potential $\lambda$V with small intensity 0<$\lambda$$\ll$1 (or
alternatively at high frequencies). The main new ingredient, which we introduce
in the general setting of a stationary ergodic potential, is an approximate
stationary Floquet--Bloch theory that is used to put the perturbed
Schr{\"o}dinger operator into approximate normal form. We apply this approach
to quasiperiodic potentials and establish a Nehoro{\v s}ev-type stability
result. In particular, this ensures asymptotic ballistic transport up to a
stretched exponential timescale exp($\lambda$--1/s) for some s>0. More
precisely, the approximate normal form leads to an accurate long-time
description of the Schr{\"o}dinger flow as an effective unitary correction of
the free flow. The approach is robust and generically applies to linear waves.
For classical waves, for instance, this allows to extend diffractive geometric
optics to quasiperiodically perturbed media.
