  We prove ballistic transport of all orders, that is, $\lVert
x^m\mathrm{e}^{-\mathrm{i}tH}\psi\rVert\asymp t^m$, for the following models:
the adjacency matrix on $\mathbb{Z}^d$, the Laplace operator on $\mathbb{R}^d$,
periodic Schr\"odinger operators on $\mathbb{R}^d$, and discrete periodic
Schr\"odinger operators on periodic graphs. In all cases we give the exact
expression of the limit of $\lVert x^m\mathrm{e}^{-\mathrm{i}tH}\psi\rVert/t^m$
as $t\to+\infty$. We then move to universal covers of finite graphs (these are
infinite trees) and prove ballistic transport in mean when the potential is
lifted naturally, giving a periodic model, and when the tree is endowed with
random i.i.d.\ potential, giving an Anderson model. The limiting distributions
are then discussed, enriching the transport theory. Some general upper bounds
are detailed in the appendix.
