  In this paper we study localization properties of the Riesz $s$-fractional
gradient $D^s u$ of a vectorial function $u$ as $s \nearrow 1$. The natural
space to work with $s$-fractional gradients is the Bessel space $H^{s,p}$ for
$0 < s < 1$ and $1 < p < \infty$. This space converges, in a precise sense, to
the Sobolev space $W^{1,p}$ when $s \nearrow 1$. We prove that the
$s$-fractional gradient $D^s u$ of a function $u$ in $W^{1,p}$ converges
strongly to the classical gradient $Du$. We also show a weak compactness result
in $W^{1,p}$ for sequences of functions $u_s$ with bounded $L^p$ norm of $D^s
u_s$ as $s \nearrow 1$. Moreover, the weak convergence of $D^s u_s$ in $L^p$
implies the weak continuity of its minors, which allows us to prove a
semicontinuity result of polyconvex functionals involving $s$-fractional
gradients defined in $H^{s,p}$ to their local counterparts defined in
$W^{1,p}$. The full $\Gamma$-convergence of the functionals is achieved only
for the case $p>n$.
