  We study the "local" behavior of several relevant properties concerning
semistar operations, like finite type, stable, spectral, e.a.b. and a.b. We
deal with the "global" problem of building a new semistar operation on a given
integral domain, by "gluing" a given homogeneous family of semistar operations
defined on a set of localizations. We apply these results for studying the
local--global behavior of the semistar Nagata ring and the semistar Kronecker
function ring. We prove that an integral domain $D$ is a Pr{\"u}fer
$\star$--multiplication domain if and only if all its localizations $D_{P}$ are
Pr{\"u}fer $\star_P$--multiplication domains.
