What are the mechanics of campaign map reinforcements on Rome Total War?

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All I can find are explanations on what causes delays with them on the battle map. I'm wondering how you get reinforcements on the campaign map. I thought it was if an army was on a tile touching the attacked city/army, they'd be brought in as a reinforcement, but when the Brutii attack Corinth with my faction leader right next to it, they're not brought in as reinforcements:
What do I have to do for my army to be brought in as reinforcements when the settlement siege starts?
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How do reinforcements work in Rome Total War?
Tap the Reinforcements button above your army to view your reinforcements, where available. If a reinforcing army belongs to your faction and is led by a General, you can uncheck the AI control box (represented by a gear icon), and that army will be under your control when you start the battle.How do you succeed in rome total war?
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