  We study the finite-time dynamics of an initially localized wave-packet in
the Anderson model on the random regular graph (RRG). Considering the full
probability distribution $\Pi(x,t)$ of a particle to be at some distance $x$
from the initial state at time $t$, we give evidence that $\Pi(x,t)$ spreads
sub-diffusively over a range of disorder strengths, wider than a putative
non-ergodic phase. We provide a detailed analysis of the propagation of
$\Pi(x,t)$ in space-time $(x,t)$ domain, identifying four different regimes.
These regimes in $(x,t)$ are determined by the position of a wave-front
$X_{\text{front}}(t)$, which moves sub-diffusively to the most distant sites
$X_{\text{front}}(t) \sim t^{\beta}$ with an exponent $\beta < 1$. We support
our numerical results by a self-consistent semiclassical picture of wavepacket
propagation relating the exponent $\beta$ with the relaxation rate of the
return probability $\Pi(0,t) \sim e^{-\Gamma t^\beta}$. Importantly, the
Anderson model on the RRG can be considered as proxy of the many-body
localization transition (MBL) on the Fock space of a generic interacting
system. In the final discussion, we outline possible implications of our
findings for MBL.
