  The swampland distance conjecture (SDC) addresses the ability of effective
field theory to describe distant points in moduli space. It is natural to ask
whether there is a local version of the SDC: is it possible to construct local
excitations in an EFT that sample extreme regions of moduli space? In many
cases such excitations exhibit horizons or instabilities, suggesting that there
are bounds on the size and structure of field excitations that can be achieved
in EFT. Static bubbles in ordinary Kaluza-Klein theory provide a simple class
of examples: the KK radius goes to zero on a smooth surface, locally probing an
infinite distance point, and the bubbles are classically unstable against
radial perturbations. However, it is also possible to stabilize KK bubbles at
the classical level by adding flux. We study the impact of imposing the Weak
Gravity Conjecture (WGC) on these solutions, finding that a rapid pair
production instability arises in the presence of charged matter with
$q/m\gtrsim 1$. We also analyze 4d electrically charged dilatonic black holes.
Small curvature at the horizon imposes a bound $\log(M_{BH})\gtrsim
|\Delta\phi|$, independent of the WGC, and the bound can be strengthened if the
particle satisfying the WGC is sufficiently light. We conjecture that quantum
gravity in asymptotically flat space requires a general bound on large
localized moduli space excursions of the form $ |\Delta\phi|\lesssim
|\log(R\Lambda)|$, where $R$ is the size of the minimal region enclosing the
excitation and $\Lambda^{-1}$ is the short-distance cutoff on local EFT. The
bound is qualitatively saturated by the dilatonic black holes and Kaluza-Klein
monopoles.
