  We study existence and convergence properties of least-energy symmetric
solutions (l.e.s.s.) to the pure critical problem \begin{equation*}
(-\Delta)^su_s=|u_s|^{2^\star_s-2}u_s, \quad u_s\in D^s_0(\Omega),\quad
2^\star_s:=\frac{2N}{N-2s}, \end{equation*} where $s$ is any positive number,
$\Omega$ is either $\mathbb{R}^N$ or a smooth symmetric bounded domain, and
$D^s_0(\Omega)$ is the homogeneous Sobolev space. Depending on the kind of
symmetry considered, solutions can be sign-changing. Once existence is
established, we show that, up to a subsequence, a l.e.s.s. $u_s$ converges to a
l.e.s.s. $u_{t}$ as $s$ goes to any $t>0$ (possibly an integer). In bounded
domains, this convergence can be characterized in terms of an homogeneous
fractional norm of order $t-\varepsilon$. A similar characterization is no
longer possible in unbounded domains due to scaling invariance and an
incompatibility with the functional spaces; to circumvent these difficulties,
we use a suitable rescaling and characterize the convergence via cut-off
functions. If $t$ is an integer, these results describe in a precise way the
nonlocal-to-local transition. Finally, we also include a nonexistence result of
nontrivial nonnegative solutions in a ball for any $s>1$.
