  We explore the quantum dynamics of particles in a spatiotemporally driven
lattice. A powerful numerical scheme is developed, which provides us with the
Floquet modes and thus enables a stroboscopic propagation of arbitrary initial
states. A detailed symmetry analysis represents the cornerstone for an
intricate manipulation of the Floquet spectrum. Specifically, we show how exact
crossings can be converted into avoided ones, while the width of these
resulting avoided crossings can be engineered by adjusting parameters of the
local driving. Asymptotic currents are shown to be controllable over a certain
parameter range.
