  Kinematically-persistent planes of satellites (KPPs) are fixed sets of
satellites co-orbiting around their host galaxy, whose orbital poles are
conserved and clustered across long cosmic time intervals. They play the role
of 'skeletons', ensuring the long-term durability of positional planes. We
explore the physical processes behind their formation in terms of the dynamics
of the local Cosmic Web (CW), characterized via the so-called Lagrangian
Volumes (LVs) built up around two zoom-in, cosmological hydro-simulations of
MW-mass disk galaxy + satellites systems, where three KPPs have been
identified. By analyzing the LVs deformations in terms of the reduced Tensor of
Inertia (TOI), we find an outstanding alignment between the LV principal
directions and KPP satellites' orbital poles. The most compressive local mass
flows (along the $\hat{e}_3$ eigenvector) are strong at early times, feeding
the so-called $\hat{e}_3$-structure, while the smallest TOI axis rapidly
decreases. The $\hat{e}_3$-structure collapse marks the end of this regime and
is the timescale for the establishment of satellite orbital pole clustering
when the Universe is $\lesssim$ 4 Gyr old. KPP proto-satellites aligned with
$\hat{e}_3$ are those whose orbital poles are either aligned from early times,
or have been successfully bent at $\hat{e}_3$-structure collapse. KPP
satellites associated to $\hat{e}_1$ tend to have early trajectories already
parallel to $\hat{e}_3$. We show that KPPs can arise as a result of the
$\Lambda$CDM-predicted large-scale dynamics acting on particular sets of
proto-satellites, the same dynamics that shape the local CW environment.
