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We review recently offered notions of quasi-static evolution of the axialself-gravitating structures at large-scales and the criteria to characterizethe corresponding evolutionary aspects under the influence of strong curvatureregimes. In doing so, we examine the axial source's dynamic and quasi-staticbehavior within the parameters of various modified gravity theories. We addressthe formalism of these notions and their possible implications in studying thedissipative and anisotropic configuration. We initiate by consideringhigher-order curvature gravity. The Palatini formalism of $f(R)$ gravity isalso taken into consideration to analyze the behavior of the kinematical aswell as the dynamical variables of the proposed problem. The set of invariantvelocities is defined to comprehend the concept of quasi-static approximationthat enhances the stability of the system in contrast to the dynamic mode. Itis identified that vorticity and distinct versions of the structure scalars$Y_{I}$, $Y_{II}$ and $Y_{KL}$ play an important role in revealing thesignificant effects of a fluid's anisotropy. As another example of evolution,we check the influence of Palatini-based factors on the shearing motion of theobject. A comparison-based study of the physical nature of distinct curvaturefactors on the propagation of the axial source is exhibited. This provides anintriguing platform to grasp the notion of quasi-static evolution together withthe distinct curvature factors at the current time scenario. The importance ofslowly evolving axially symmetric regimes will be addressed through thedistinct modified gravitational context. Finally, we share a list of queriesthat, we believe, deserve to be addressed in the near future.

Quasi-static evolution of axially and reflection symmetric large-scale configuration