In general Yang-Mills theory  seems to be different from the dimensional reduced Kaluza–Klein theory.
However, the historical account was that people tried to trace back the origin of non-Abelian gauge theory (generalize the 2-form field strength / curvature for 1-form gauge field connection from the fiber bundle of non-abelian Lie group, such as SU(n)) even before the works of Yang-Mills in 1954 (Phys. Rev. 96, 191 – Published 1 October 1954).
In a private correspondence, Wolfgang Pauli formulated in 1953 a six-dimensional theory of Einstein's field equations of general relativity, extending the five-dimensional theory of Kaluza, Klein, Fock and others to a higher-dimensional internal space. However, there is no evidence that Pauli developed the Lagrangian of a gauge field or the quantization of it. Because Pauli found that his theory "leads to some rather unphysical shadow particles", he refrained from publishing his results formally. Although Pauli did not publish his six-dimensional theory, he gave two talks about it in Zürich.
In Wikipedia, it says that "Recent research shows that an extended Kaluza–Klein theory is in general not equivalent to Yang–Mills theory, as the former contains additional terms "
But if one tries to look at the Ref , this Ref seems not be very accessible (in the sense that neither the physics nor the math statements are given in a standard form --- the Ref  is unpublished and type-set in the Microsoft Word format.
My question is: Are there some math literatures fulfilling the similar logical gap to compare the Yang-Mills theory v.s. dimensional reduced Kaluza–Klein theory? The general conditions for equivalence or inequivalence for the classical actions?
Yang, C. N.; Mills, R. (1954). "Conservation of Isotopic Spin and Isotopic Gauge Invariance". Physical Review. 96 (1): 191–195
Straumann, N (2000). "On Pauli's invention of non-abelian Kaluza-Klein Theory in 1953". arXiv:gr-qc/0012054
Reifler, N (2007). "Conditions for exact equivalence of Kaluza-Klein and Yang–Mills theories". arXiv:0707.3790