  In this work, we analyze the coset space dimensional reduction scheme to
construct pure Yang--Mills fields on maximally symmetric spacetimes given as
cylinders over cosets. Particular cases of foliations using $H^n$, dS$_n$ and
AdS$_n$ slices as non-compact symmetric spaces are solved, compared to previous
results in the literature, and generalized in a structured fashion. Coupling to
General Relativity in FLRW-type universes is introduced via the cosmological
scale factor. For the hyperbolic slicing, the dynamics of the
Einstein--Yang--Mills system is analytically solved and discussed. Finally, we
generalize the analysis to warped foliations of the cylinders, which enlarge
the range of possible spacetimes while also introducing a Hubble friction-like
term in the equation of motion for the Yang--Mills field.
