  The Cobordism Conjecture states that any Quantum Gravity configuration
admits, at topological level, a boundary ending spacetime. We study the
dynamical realization of cobordism, as spacetime dependent solutions of
Einstein gravity coupled to scalars containing such end-of-the-world "branes".
The latter appear in effective theory as a singularity at finite spacetime
distance at which scalars go off to infinite field space distance. We provide a
local description near the end-of-the-world branes, in which the solutions
simplify dramatically and are characterized in terms of a critical exponent,
which controls the asymptotic profiles of fields and the universal scaling
relations among the spacetime distance to the singularity, the field space
distance, and the spacetime curvature. The analysis does not rely on
supersymmetry. We study many explicit examples of such Local Dynamical
Cobordisms in string theory, including 10d massive IIA, the 10d
non-supersymmetric $USp(32)$ theory, Bubbles of Nothing, 4d $ \mathcal{N}=1 $
cosmic string solutions, the Klebanov-Strassler throat, D$p$-brane solutions,
brane configurations related to the D1/D5 systems, and small black holes. Our
framework encompasses diverse recent setups in which scalars diverge at the
core of defects, by regarding them as suitable end-of-the-world branes. We
explore the interplay of Local Dynamical Cobordisms with the Distance
Conjecture and other swampland constraints.
