  The generalized Franchetta conjecture for hyper-K\"ahler varieties predicts
that an algebraic cycle on the universal family of certain polarized
hyper-K\"ahler varieties is fiberwise rationally equivalent to zero if and only
if it vanishes in cohomology fiberwise. We establish Franchetta-type results
for certain low (Hilbert) powers of low degree K3 surfaces, for the
Beauville--Donagi family of Fano varieties of lines on cubic fourfolds and its
relative square, and for $0$-cycles and codimension-$2$ cycles for the
Lehn--Lehn--Sorger--van Straten family of hyper-K\"ahler eightfolds. We also
draw many consequences in the direction of the Beauville--Voisin conjecture as
well as Voisin's refinement involving coisotropic subvarieties. In the
appendix, we establish a new relation among tautological cycles on the square
of the Fano variety of lines of a smooth cubic fourfold and provide some
applications.

Algebraic Geometry