  We study the local dynamical fluctuations in glass-forming models of
particles embedded in $d$-dimensional space, in the mean-field limit of
$d\to\infty$. Our analytical calculation reveals that single-particle
observables, such as squared particle displacements, display divergent
fluctuations around the dynamical (or mode-coupling) transition, due to the
emergence of nontrivial correlations between displacements along different
directions. This effect notably gives rise to a divergent non-Gaussian
parameter, $\alpha_2$. The $d\to\infty$ local dynamics therefore becomes quite
rich upon approaching the glass transition. The finite-$d$ remnant of this
phenomenon further provides a long sought-after, first-principle explanation
for the growth of $\alpha_2$ around the glass transition that is \emph{not
based on multi-particle correlations}.
