Added the bit on reinforcing generals earning gravitas. This might be one of the reasons why many people claim their party's generals win battles yet the party does not gain influence:

Generals joining the battle as reinforcements earn gravitas too. This is something I noticed recently. The generals who join your battle as reinforcements do not earn battle experience. However, they DO earn gravitas. Thus, if your main general wins the battle and earns gravitas, yet your reinforcing army was led by an opposition general who also gained gravitas, the total impact on your party's influence could be nil (both party's earned +1 gravitas creating a wash situation unless the ambition differential of generals skews it one way or another). So, better have your own party's general leading the reinforcing army if you have one.