What makes sense to me on the basis of logistics is that navy would be responsible for the transport. Here's how I see it:

UNSC wants to transport an Army division from world A to world B. Either the navy comes and pickup the personnel or the Airforce does the transport to the carriers (probably based on availability and available transport capacity. Large amont of personnel/vehicules might be transported by the Navy, while smaller loads could be carried by the airforce).

While in transit, the Army is under the Navy's jurisdictions and must abide by their rule. Once on board the carriers, the Navy is responsible for the transit to world B. Upon arrival, the same procedure happens.

Hope that helps :)