  We study the large field limit in Schr\"odinger equations with magnetic
vector potentials describing translationally invariant $B$-fields with respect
to the $z$-axis. In a first step, using regular perturbation theory, we derive
an approximate description of the solution, provided the initial data is
compactly supported in the Fourier-variable dual to $z\in \mathbb R$. The
effective dynamics is thereby seen to produce high-frequency oscillations and
large magnetic drifts. In a second step we show, by using the theory of almost
invariant subspaces, that this asymptotic description is stable under
polynomially bounded perturbations that vanish in the vicinity of the origin.
