  Gushel-Mukai sixfolds are an important class of so-called Fano-K3 varieties.
In this paper we show that they admit a multiplicative Chow-K\"unneth
decomposition modulo algebraic equivalence and that they have the Franchetta
property. As side results, we show that double EPW sextics and cubes have the
Franchetta property, modulo algebraic equivalence, and some vanishing results
for the Chow ring of Gushel-Mukai sixfolds.
