  Strong anomalous diffusion is characterized by asymptotic power-law growth of
the moments of displacement, with exponents that do not depend linearly on the
order of the moment. The exponents concerning small-order moments are dominated
by random motion, while higher-order exponents grow by faster trajectories,
such as ballistic excursions or "light fronts". Often such a situation is
characterized by two linear dependencies of the exponents on their order. Here,
we introduce a simple exactly solvable model, the Fly-and-Die (FnD) model, that
sheds light on this behavior and on the consequences of light fronts on
displacement autocorrelation functions in transport processes. We present
analytical expressions for the moments and derive a scaling form that expresses
the long-time asymptotics of the autocorrelation function $\langle
x(t_1)\,x(t_2)\rangle$ in terms of the dimensionless time difference
$(t_2-t_1)/t_1$. The scaling form provides a faithful collapse of numerical
data for vastly different systems. This is demonstrated here for the Lorentz
gas with infinite horizon, polygonal billiards with finite and infinite
horizon, the L\'evy-Lorentz gas, the Slicer Map, and L\'evy walks. Our analysis
also captures the system-specific corrections to scaling.
