Logical transformations provide a very convenient tool for encoding classes of structures within other classes of structures, and some important class properties can be defined in terms of transformations. In this paper, we study first-order (FO) transformations and the sub-degrees they lead to infinite classes of finite graphs. Surprisingly, this quasi-order is highly complex, but shaped by the locality of first-order logic. This contrasts with the expected simplicity of the unary quadratic (MSO) transform suborder. First, we establish the local normal form of the FO transform, which is of independent concern. This canonical form allows us to prove that the local variant of (monadic) stability and (monadic) dependencies is equivalent to the non-local version. We then prove that the quotient partial order is a bounded partition-coupling semilattice, and that the additive class subset is also a bounded partition-coupling semilattice. Characterize transformations of paths, cubic graphs, and cubic trees in terms of bandwidth, bounded degree, and tree width. For $k\geq 1$, the classes of all graphs with path width at most $k$ form a strict hierarchy in the FO transform semi-order, and the same is true for all graphs with tree width at most $k$. It reveals whether it holds for the class of k$. It identifies the obstacle that the class is a transformation of a class with bounded degree, leading to an interesting transformation duality formulation. Finally, we discuss the notion of dense analogues of sparse transformation-preserving class properties and propose some related conjectures.
