  The purpose of this short note is to study dominant rational maps from
punctual Hilbert schemes of length $k>1$ of projective K3 surfaces $S$
containing infinitely many rational curves. Precisely, we prove that their
image is necessarily rationally connected if this rational map is not
generically finite. As an application, we simplify the proof of C. Voisin's of
the fact that symplectic involutions of any projective K3 surface $S$ act
trivially on $\mathrm{CH}_0(S)$.
