We provide analogues of the results from Friedman and Motto Ros (2011) and Camerlo, Marcone, and Motto Ros (2013) (which correspond to the case
κ
=
ω
\kappa = \omega
) for arbitrary
κ
\kappa
-Souslin quasi-orders on any Polish space, for
κ
\kappa
an infinite cardinal smaller than the cardinality of
R
\mathbb {R}
. These generalizations yield a variety of results concerning the complexity of the embeddability relation between graphs or lattices of size
κ
\kappa
, the isometric embeddability relation between complete metric spaces of density character
κ
\kappa
, and the linear isometric embeddability relation between (real or complex) Banach spaces of density
κ
\kappa
.