Does anyone know what happens if one is a European citizen (or dual-citizen with the US) but physically resident in the US? Can one invoke EU rights, or is it

Does anyone know what happens if one is a European citizen (or dual-citizen with the US) but physically resident in the US? Can one invoke EU rights, or is it tied to being physically resident like a DVD region? What if the EU citizen flies to Europe? Can they then invoke their right from the physical territory of Europe? I feel this is somewhat uncharted water.