  In this paper we describe a well-chosen discrete moving frame and their
associated invariants along projective polygons in $\RP^n$, and we use them to
write explicit general expressions for invariant evolutions of projective
$N$-gons. We then use a reduction process inspired by a discrete
Drinfeld-Sokolov reduction to obtain a natural Hamiltonian structure on the
space of projective invariants, and we establish a close relationship between
the projective $N$-gon evolutions and the Hamiltonian evolutions on the
invariants of the flow. We prove that {any} Hamiltonian evolution is induced on
invariants by an evolution of $N$-gons - what we call a projective realization
- and we give the direct connection. Finally, in the planar case we provide
completely integrable evolutions (the Boussinesq lattice related to the lattice
$W_3$-algebra), their projective realizations and their Hamiltonian pencil. We
generalize both structures to $n$-dimensions and we prove that they are
Poisson. We define explicitly the $n$-dimensional generalization of the planar
evolution (the discretization of the $W_n$-algebra) and prove that it is
completely integrable, providing also its projective realization.
