  In this paper, we present a framework to construct sequences of spectral
triples on top of an inductive sequence defining an $AF$-algebra. One aim of
this paper is to lift arrows of a Bratteli diagram to arrows between Krajewski
diagrams. The spectral actions defining Non-commutative Gauge Field Theories
associated to two spectral triples related by these arrows are compared
(tensored by a commutative spectral triple to put us in the context of Almost
Commutative manifolds). This paper is a follow up of a previous one in which
this program was defined and physically illustrated in the framework of the
derivation-based differential calculus, but the present paper focuses more on
the mathematical structure without trying to study the physical implications.
