  Consider the family of Schr\"odinger operators (and also its Dirac version)
on $\ell^2(\mathbb{Z})$ or $\ell^2(\mathbb{N})$ \[ H^W_{\omega,S}=\Delta +
\lambda F(S^n\omega) + W, \quad \omega\in\Omega, \] where $S$ is a
transformation on (compact metric) $\Omega$, $F$ a real Lipschitz function and
$W$ a (sufficiently fast) power-decaying perturbation. Under certain conditions
it is shown that $H^W_{\omega,S}$ presents quasi-ballistic dynamics for
$\omega$ in a dense $G_{\delta}$ set. Applications include potentials generated
by rotations of the torus with analytic condition on $F$, doubling map, Axiom A
dynamical systems and the Anderson model. If $W$ is a rank one perturbation,
examples of $H^W_{\omega,S}$ with quasi-ballistic dynamics and point spectrum
are also presented.
