We consider a particle moving in $d\geq 2$ dimensions, its velocity being a reversible diffusion process, with identity diffusion coefficient, of which the invariant measure behaves, roughly, like $(1+|v|)^{-β}$ as $|v|\to \infty$, for some constant $β>0$. We prove that for large times, after a suitable rescaling, the position process resembles a Brownian motion if $β\geq 4+d$, a stable process if $β\in [d,4+d)$ and an integrated multi-dimensional generalization of a Bessel process if $β\in (d-2,d)$. The critical cases $β=d$, $β=1+d$ and $β=4+d$ require special rescalings.