  We review recently offered notions of quasi-static evolution of the axial
self-gravitating structures at large-scales and the criteria to characterize
the corresponding evolutionary aspects under the influence of strong curvature
regimes. In doing so, we examine the axial source's dynamic and quasi-static
behavior within the parameters of various modified gravity theories. We address
the formalism of these notions and their possible implications in studying the
dissipative and anisotropic configuration. We initiate by considering
higher-order curvature gravity. The Palatini formalism of $f(R)$ gravity is
also taken into consideration to analyze the behavior of the kinematical as
well as the dynamical variables of the proposed problem. The set of invariant
velocities is defined to comprehend the concept of quasi-static approximation
that enhances the stability of the system in contrast to the dynamic mode. It
is identified that vorticity and distinct versions of the structure scalars
$Y_{I}$, $Y_{II}$ and $Y_{KL}$ play an important role in revealing the
significant effects of a fluid's anisotropy. As another example of evolution,
we check the influence of Palatini-based factors on the shearing motion of the
object. A comparison-based study of the physical nature of distinct curvature
factors on the propagation of the axial source is exhibited. This provides an
intriguing platform to grasp the notion of quasi-static evolution together with
the distinct curvature factors at the current time scenario. The importance of
slowly evolving axially symmetric regimes will be addressed through the
distinct modified gravitational context. Finally, we share a list of queries
that, we believe, deserve to be addressed in the near future.
