  We prove a Carath\'eodory-type extension of BQS homeomorphisms between two
domains in proper, locally path-connected metric spaces as homeomorphisms
between their prime end closures. We also give a Carath\'eodory-type extension
of geometric quasiconformal mappings between two such domains provided the two
domains are both Ahlfors $Q$-regular and support a $Q$-Poincar\'e inequality
when equipped with their respective Mazurkiewicz metrics. We also provide
examples to demonstrate the strengths and weaknesses of prime end closures in
this context.
