The Russians had more immediate ammunition than the French so could fight their gun independent of their caissons. The Russian Caisson design was superior to the Gribeauval design that was in effect a wagon with small front wheels. The larger wheel of the two wheeled Russian Ammunition Caisson made it better for the poor roads in Russia and could be hauled by three horses abrest so having a higher traction. Being half the weight also assisted in the cross country ability.
Most nations used fixed ammunition where the shot was attached via the wooden sabot to the gunpowder charge. The gunpowder cartridge was made of serge or paper. I am not sure as to the construction of Russian Cartridges.
The important part was that the caisson was water-tight. The Gribeauval design suffered from being less than water-tight. The design of the AnXI Caisson with ammunition boxes as used by other countries where each box was waterproofed was superior to the individual placing within divisions that was used by the Russians and Gribeauval.
It is interesting looking at the ammunition distribution within a caisson to understand that only the Howitzers were normally equiped with separate powder charges. Range for guns were determined by the elevation and that for howitzers by the powder charge size with a lesser extent to elevation.