  Let $X$ be a very general Gushel-Mukai (GM) variety of dimension $n\geq 4$,
and let $Y$ be a smooth hyperplane section. There are natural pull-back and
push-forward functors between the semi-orthogonal components (known as the
Kuznetsov components) of the derived categories of $X$ and $Y$. In the first
part of the paper, we prove that the Bridgeland stability of objects is
preserved by both pull-back and push-forward functors. We then explore various
applications of this result, such as constructing an 8-dimensional smooth
family of Lagrangian subvarieties for each moduli space of stable objects in
the Kuznetsov component of a general GM fourfold, and proving the projectivity
of the moduli spaces of semistable objects of any class in the Kuznetsov
component of a general GM threefold, as conjectured by Perry, Pertusi, and
Zhao.
  In the second part, we study in detail the double EPW cube, a 6-dimensional
hyperk\"ahler manifold associated with a very general GM fourfold $X$, through
the Bridgeland moduli space, and show that it is the maximal rationally
connected (MRC) quotient of the Hilbert scheme of twisted cubics on $X$. We
also prove that it admits a covering by Lagrangian subvarieties, constructed
from the Hilbert schemes of twisted cubics on GM threefolds.
