Thoughts on an initiative system, to be used with any game that uses initiative.
First, we assert that the game has a designation between combat and non-combat, and that normally, combat is started by rolling initiative. In this system, instead, only the first actor acts; everyone else reacts.
Combat begins when someone in the fiction takes an action that would start combat. Stabbing someone, casting an offensive spell, firing a crossbow, glove slap, whatever action is agreed that it starts a fight. This character takes their action first. If you want to act first, then you should declare your fight-starting sooner. (Does this incentivize players to be sociopaths? Yes. But it's also the same incentives that reality has, so it also incentivizes things like not letting diplomats wear weapons. Except for an old dwarf wizard's walking axe, of course.)
All actions that characters can take have triggers. These actions can only be used when an actor triggers them. The most basic triggered action would be Counterattack, which lets you make an attack when you are attacked. More complex ones may include things like; use when fire damage is dealt, use when an ally is attacked, use when an enemy is reduced to 0 HP, etc.
Actions cost action points. Characters have some number of action points determined by their class, level, or other appropriate stats (in Cyberpunk 2020/RED, for example, you'd probably get action points based on your REF).
When an action is taken, it's reasonably likely that more than one character in the battle will have a reaction to it. Priority, in general, is given to actions with more action points spent on them. You can spend extra action points to raise your priority if you want. There are two ideas here to resolve multiple characters reacting.
1) Only the character who bids the most action points can take their desired action. This is called seizing the initiative. Only the character who seizes the initiative actually spends their action points, everyone who loses gets to keep theirs.
2) Everyone who has an action triggered can queue their action up. The actions are resolved in order of action points spent, from most to least. A triggering action and all reactions to it are referred to as an action cloud. No one can take any further action until the action cloud is fully resolved, but then they can react to anything that happened as part of the action cloud.
When no one in the combat has any more actions they want to take (or are able to take), everyone gets their action points back. If everyone has full action points, and no one is able to (or chooses not to) take an action, combat ends. This may result in combat starting again after a brief interlude, if combat ended because no one had an available triggered action.
The primary challenges in turning this system from a concept into a mechanic would be
1) Defining available action points and costs. This is a nontrivial grid-filling task, depending on system, because you would need to define this for all available actions.
2) Defining triggers for actions. As above, this might be a lot of work, depending on the system. For some systems, everything is covered by a set of basic actions that would make this relatively easy. If you decide to define a separate action point cost and trigger for every spell in a D&D-like game, I wish you good luck.