  The cobordism conjecture implies that consistent theories of Quantum Gravity
must admit the introduction of boundaries. We study the dynamical realization
of the cobordism conjecture in type IIB in AdS$_5\times \bf{S}^5$, using the
existing gravity duals of 4d $\mathcal{N}=4$ SYM with Gaiotto-Witten
superconformal boundary conditions (near-horizon limits of D3-branes ending on
NS5- and D5-branes). We show that these configurations are, from the 5d
perspective, dynamical cobordism solutions which start from an asymptotic
AdS$_5$ vacuum and evolve until they hit an end of the world (ETW) brane with
AdS$_4$ worldvolume. The latter displays localization of gravity, and provide a
completion of the Karch-Randall (KR) AdS branes, in which the backreaction of
running scalars replace the KR cusp in the warp factor with a smooth bump. The
dynamical scalars are either in the $SO(6)$ invariant AdS$_5$ bulk sector (e.g.
describing the $\bf{S}^5$ size and its shrinking at the cobordism boundary) or
brane localized (e.g. the $SO(6)\to SO(3)\times SO(3)$ squashing due to
boundary conditions). We introduce a novel double scaling limit which zooms
into the ETW brane and makes localization of gravity manifest, and which shows
a tantalizing relation with wedge holography. We extend the picture to AdS$_5$
theories with less (super)symmetry, via orbifolds and S-folds, leading to
dynamical cobordisms for gravity duals of 4d theories with $\mathcal{N}=2$ and
$\mathcal{N}=3$ supersymmetry.
