Regarding K3 surfaces in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$, you might find the following two articles relevant. (Also forward trace them as references for further articles.)

1. Baragar, A. Rational points on K3 surfaces in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$. Math. Ann. 305 (1996), no. 3, 541-558. (MR1397435)

2. Wang, Lan Rational points and canonical heights on K3-surfaces in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$. Recent developments in the inverse Galois problem (Seattle, WA, 1993), 273-289, Contemp. Math., 186, Amer. Math. Soc., Providence, RI, 1995 (MR1352278)

Regarding K3 surfaces in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$, you might find the following two articles relevant. (Also forward trace them as references for further articles.)

1. Baragar, A. Rational points on K3 surfaces in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$. Math. Ann. 305 (1996), no. 3, 541-558. (MR1397435)

2. Wang, Lan Rational points and canonical heights on K3-surfaces in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$. Recent developments in the inverse Galois problem (Seattle, WA, 1993), 273-289, Contemp. Math., 186, Amer. Math. Soc., Providence, RI, 1995 (MR1352278)

Addendum Since you mention that you're interested in higher dimensional analogous, I'll mention the following. The involutions on a $(2,2,2)$-surface in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$ are morphisms. But if $n\ge4$, then the involutions on a $(2,2,\ldots,2)$ hypersurface in $\underbrace{\mathbb P^1\times\mathbb P^1\times\cdots\times\mathbb P^1}_{\text{$n$ factors}}$ have non-trivial indeterminacy locus, so some points have orbits that terminate.