Let κ ≥ 2 be a fixed natural number. The complete description is given of the product preserving gauge bundle functors F on the category F κ VB of flag vector bundles K = (K; K 1 ,. .. , K κ) of length κ in terms of the systems I = (I 1 ,. .. , I κ−1) of A-module homomorphisms I i : V i+1 → V i for Weil algebras A and finite dimensional (over R) A-modules V 1 ,. .. , V κ. The so called iteration problem is investigated. The natural affinors on FK are classified. The gauge-natural operators C lifting κ-flag-linear (i.e. with the flow in F κ VB) vector fields X on K to vector fields C(X) on FK are completely described. The concept of the complete lift F φ of a κ-flag-linear semi-basic tangent valued p-form φ on K is introduced. That the complete lift F φ preserves the Frölicher-Nijenhuis bracket is deduced. The obtained results are applied to study prolongation and torsion of κ-flag-linear connections.