  We study the variation of the local Langlands correspondence for ${\rm
GL}_{n}$ in characteristic-zero families. We establish an existence and
uniqueness theorem for a correspondence in families, as well as a recognition
theorem for when a given pair of Galois- and reductive-group- representations
can be identified as local Langlands correspondents. The results, which can be
used to study local-global compatibility questions along eigenvarieties, are
largely analogous to those of Emerton, Helm, and Moss on the local Langlands
correspondence over local rings of mixed characteristic. We apply the theory to
the interpolation of local zeta integrals and of $L$- and
$\varepsilon$-factors.
