There is no carryover in a phase-based system with no rounds.
Without rounds, your agility or dex might lower the action point cost for certain things. Say swinging a sword takes 10 APs.
Your Dex or Agi might give as much as a 1 through 5 modifier to APs, negative or positive.
So with a super high Agility, you can swing a sword in half the time.
For initiative, you roll a D10, and that says how far in the phase you start at. A good init roll allows you to get a quick jump.
You have a Agi modifier of 3, so it takes a 7 for you to swing a sword. You roll a 5 on initiative, so that's where you start. You effectively start combat with 5 free action points removed from your first action. You won't actually get to swing your sword into 2 phases or action points into battle. And then you swing again 7 more phases in.
You're effectively swinging on 2, 9, 16, 23, so on and so forth. But if you choose another action down the round, your next action could be sooner or later based on the number of action points for said action.