We present a detailed analysis of the local evolution of 206 Lagrangian Volumes (LVs) selected at high redshift around galaxy seeds, identified in a large-volume $Lambda$ cold dark matter ($Lambda$CDM) hydrodynamical simulation. The LVs have a mass range of $1 - 1500 times 10^{10} M_odot$. We follow the dynamical evolution of the density field inside these initially spherical LVs from $z=10$ up to $z_{rm low} = 0.05$, witnessing highly non-linear, anisotropic mass rearrangements within them, leading to the emergence of the local cosmic web (CW). These mass arrangements have been analysed in terms of the reduced inertia tensor $I_{ij}^r$, focusing on the evolution of the principal axes of inertia and their corresponding eigendirections, and paying particular attention to the times when the evolution of these two structural elements declines. In addition, mass and component effects along this process have also been investigated. We have found that deformations are led by dark matter dynamics and they transform most of the initially spherical LVs into prolate shapes, i.e. filamentary structures. An analysis of the individual freezing-out time distributions for shapes and eigendirections shows that first most of the LVs fix their three axes of symmetry (like a skeleton) early on, while accretion flows towards them still continue. Very remarkably, we have found that more massive LVs fix their skeleton earlier on than less massive ones. We briefly discuss the astrophysical implications our findings could have, including the galaxy mass-morphology relation and the effects on the galaxy-galaxy merger parameter space, among others.
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