  Anomalous dynamics in which local perturbations spread faster than diffusion
are ubiquitously observed in the long-time behavior of a wide variety of
systems. Here, the manner by which such systems evolve towards their asymptotic
superdiffusive behavior is explored using the 1d L\'evy walk of order $1 <
\beta < 2$. The approach towards superdiffusion, as captured by the leading
correction to the asymptotic behavior, is shown to remarkably undergo a
transition as $\beta$ crosses the critical value $\beta_{c} = 3/2$. Above
$\beta_{c}$, this correction scales as $\lvert x \rvert \sim t^{1/2}$,
describing simple diffusion. However, below $\beta_{c}$ it is instead found to
remain superdiffusive, scaling as $\lvert x \rvert \sim t^{1/(2\beta-1)}$. This
transition is shown to be independent of the precise model details and is thus
argued to be universal.
