  In this paper, we define canonical lifts of vector fields to the
multisymplectic multimomentum bundles of De Donder-Weyl Hamiltonian first-order
field theories and to the appropriate premultisymplectic embedded constraint
submanifolds on which singular field theories are studied. These new canonical
lifts are used to study the so-called natural Noether symmetries present in
both regular and singular Hamiltonian field theories along with their
associated conserved quantities obtained from Noether's theorem. The
Klein-Gordon field, the Polyakov bosonic string, and Einstein-Cartan gravity in
3 + 1 dimensions are analyzed in depth as applications of these concepts; as a
peripheral result obtained in the analysis of the bosonic string, we provide a
new geometrical interpretation of the well-known Virasoro constraint.
