From my Reserve Infantry days back in 1978-1982, we had one cleaning kit pers section; I think the C2 Gunner carried it. Every rifleman, including the C2 Gunner, could keep his weapon clean with the Oil Bottle and Pull-Through which was carried in the Butt and a few 4x2s carried in the Grenade Pouch or suitable pocket. The C1 SMG had its own kit that the Section Commander carried. The section Cleaning Kit came out when everyone was giving their weapons a good cleaning.
The GPMG had its own Cleaning Kit, some of which were from WWII.
SMG Cleaning Kit
FN C1A1 - C2 Cleaning Kit
C6 M1919A4 GPMG Cleaning Kit