  We investigate the nonequilibrium dynamics of classical particles in a driven
quasiperiodic lattice based on the Fibonacci sequence. An intricate transient
dynamics of extraordinarily long ballistic flights at distinct velocities is
found. We argue how these transients are caused and can be under- stood by a
hierarchy of block decompositions of the quasiperiodic lattice. A comparison to
the cases of periodic and fully randomized lattices is performed.
