  We investigate the behavior of the Weak Gravity Conjecture (WGC) under
toroidal compactification and RG flows, finding evidence that WGC bounds for
single photons become weaker in the infrared. By contrast, we find that a
photon satisfying the WGC will not necessarily satisfy it after toroidal
compactification when black holes charged under the Kaluza-Klein photons are
considered. Doing so either requires an infinite number of states of different
charges to satisfy the WGC in the original theory or a restriction on allowed
compactification radii. These subtleties suggest that if the Weak Gravity
Conjecture is true, we must seek a stronger form of the conjecture that is
robust under compactification. We propose a "Lattice Weak Gravity Conjecture"
that meets this requirement: a superextremal particle should exist for every
charge in the charge lattice. The perturbative heterotic string satisfies this
conjecture. We also use compactification to explore the extent to which the WGC
applies to axions. We argue that gravitational instanton solutions in theories
of axions coupled to dilaton-like fields are analogous to extremal black holes,
motivating a WGC for axions. This is further supported by a match between the
instanton action and that of wrapped black branes in a higher-dimensional UV
completion.
