  We study base-point-freeness for big and nef line bundles on hyperk\"ahler
manifolds of generalized Kummer type: For $n\in \{2,3,4\}$, we show that,
generically in all but a finite number of irreducible components of the moduli
space of polarized $\mathrm{Kum}^n$-type varieties, the polarization is
base-point-free. We also prove generic base-point-freeness in the moduli space
in all dimensions if the polarization has divisibility one.
