  We investigate, using the density matrix renormalization group, the evolution
of the Nagaoka state with $t'$ hoppings that frustrate the hole kinetic energy
in the $U=\infty$ Hubbard model on the anisotropic triangular lattice and the
square lattice with second-nearest neighbor hoppings. We find that the Nagaoka
ferromagnet survives up to a rather small $t'_c/t \sim 0.2.$ At this critical
value, there is a transition to an antiferromagnetic phase, that depends on the
lattice: a ${\bf Q}=(Q,0)$ spiral order, that continuously evolves with $t'$,
for the triangular lattice, and the usual ${\bf Q}=(\pi,\pi)$ N\'eel order for
the square lattice. Remarkably, the local magnetization takes its classical
value for all considered $t'$ ($t'/t \le 1$). Our results show that the
recently found classical kinetic antiferromagnetism, a perfect counterpart of
Nagaoka ferromagnetism, is a generic phenomenon in these kinetically frustrated
electronic systems.
