  We prove that the Kodaira dimension of the n-fold universal family of
lattice-polarized holomorphic symplectic varieties with dominant and
generically finite period map stabilizes to the moduli number when n is
sufficiently large. Then we study the transition of Kodaira dimension
explicitly, from negative to nonnegative, for known explicit families of
polarized symplectic varieties. In particular, we determine the exact
transition point in the cases of Beauville-Donagi and Debarre-Voisin, where the
Borcherds Phi_{12} form plays a crucial role.
