[thejeff]

Battle resolution is independent in each phase.
Obviously fortress battles depend on the outcome of regular battles which depend on the outcome of ritual phase battles, but otherwise they don't affect each other.

The outcome of battles will affect where routed troops go, but that's handled after they are all resolved anyway. Troops routed from the first battle don't show up in a later battle in an adjacent province. They're spread out among any neighboring friendly provinces after all battles are resolved.