For operations in and from space, you will not be "sending in the marines." For Cislunar forces the exact terminology we will use will be "Space Assault Forces." I am basing it on the prototype of the Russian Vozdushno-Desantnye Voiska mostly because this is a distinct arm of service within the Russian military. Since the forces would originate from the contributing polities, each with its own traditions, individual formations may hold that their transitions originate with Marine, Airborne, or Army units; but new traditions would doubtless form rapidly.
In this model, the mission of the SAF would center around:
- Operations involving planetary assault or raiding from orbit against primarily human-habitable targets.
- Conduct of planetary operations on habitable but unearth-like environments. This makes adaptability and flexibility a key requirement. SAF aircraft, for example, would have to be configurable for multiple gravities and atmospheric pressure profiles; and adaptability that would probably come at a cost compared to craft optimized for the environment.
- SAF would probably also have formations (and elements within larger forces) that are true space combat specialists, concentrating on shipboard, zero-gravity, and domed-colony assault work. These would probably also form teams for ship security and boarding actions.
Neither, for that matter will search and rescue, or support for civil authorities. The exact combination of forces that will perform such roles will depend on the polity. Many states have certain military (like the US National Guard) or paramilitary (the French Gendarmerie nationale, the Internal Troops of the Ministry of Internal Affairs of the Russian Federation) forces that perform more civil-directed action. Which a given polity or colony world might use would depend on population and on the traditions of the main founding polities.
The space-based polities - Jupiter and Saturn and their subject objects - would probably not have extensive ground troops, while individual combat specialists would be purely of the space-based types. Large domes, like Luna Farside, would need internal security troops, but these would be more of the Paramilitary Police type.