The vortex equations describe BPS configurations in gauged sigma-models on surfaces. Their moduli spaces support Kähler metrics encoding information about the underlying field theories, at both classical and quantum level. An interesting setting is when the target of these field theories is nonlinear -- i.e. a Kähler manifold with holomorphic and Hamiltonian action which does not correspond to a group representation. This setup gives rise to interesting phenomena that are not present in more familiar field theory models that it interpolates, namely, the sigma-model (trivial group) and the gauged linear sigma-model (linear action). Examples of such phenomena are the coexistence of more than one type of solitonic "particle" within the same BPS configuration, and the emergence of boundaries on the moduli spaces that correspond to coalescence of different BPS particles. In my talk, I will report on joint work with Martin Speight on global geometry and asymptotics of the moduli space metrics, focusing on the case where the target is the 2-sphere with usual circle action.