So I've thought about this and I've considered a lot of the feedback I've received. I appreciate much of the feedback regarding the various initiative systems that people utilize to make their combats feel more dynamic. However, I still feel like each of those systems leaves out of the equation more concrete ways or opportunities for players to help each other out or combine their efforts in ways that create new results. I also agree with many of the critics about creating a new action type, as it can have wide ranging impacts on action economy. Suddenly there is the question if monsters and NPCs can do these actions, and how would this complicate/slow down combat.
And so I've decided to make each of these use a player's reaction. This has the benefit of increasing options for reactions (which are rather limited without feats or special abilities), while maintaining current action economy.
New Types of Reactions
Battlefield Mobility: As a reaction, at the end of an enemy's turn you can move up to your speed. If you have already gone this round and have any leftover movement from your turn, you can also use it during this reaction. This movement may draw attacks of opportunity as normal.
Save Yourselves! (Name needs work): You can position yourself to assist your ally to help them get out of combat. When an ally attempts to move out of an enemy's reach, you can use your reaction move up to your speed and make an opposed Athletics or Acrobatics check against that enemy if they are within 5' of you. The enemy gets to choose if they will use Athletics or Acrobatics. If you succeed, that enemy cannot make attacks of opportunity until the start of your next turn.
Running Interference: You can move with an ally and provide cover for them. When an ally that is within 5' begins moving, you can move with them. Moving with your ally in this way allows you to take any attacks of opportunity that would normally be directed at your ally. If you move more than your base speed during this action, you take one level of exhaustion for every multiple of your base movement speed you cover in total distance (i.e. 1 level of exhaustion if you move more than your base speed, 2 levels of exhaustion if you move more than twice your base speed, ect).
Class Specific Battlefield Actions
Barbarian
Tactical Meat Shield (Name needs work): When an ally is hit with an attack or fails a saving throw that specifically targets them, you can use your reaction to move up to your speed towards them. If you are within 5' of your ally by the end of this movement, you can force yourself in the path of the attack or effect. You become the target of the effect and automatically take any damage and are subject to any effects resulting from the attack.
Reasoning: Barbarians are usually the toughest hombres around. They can take blows that others couldn't survive. Also, until higher levels, you need to be receiving damage or making attacks to sustain your rage. This provides one more way to make sure your rage does not end prematurely, especially if you are facing foes at a range.
Bard
Tactical Spell Enchantment: When an ally within 60' of you casts a spell or uses an ability that targets an enemy, as a reaction you can use one of your spell slots to make your ally's spell or ability more likely to take effect. Divide the level of the spell slot you sacrificed by two (round up). You add this value to your ally’s DC or Attack Bonus.
Reasoning: Bards are all about party support anyways, and there is some question whether Bards should even be full casters. This gives Bards a new way to use their spells to support the team without having to only rely on Bardic Inspiration. Also, this mechanic seems to be inline with Bardic Inspiration, at least in spirit.
Cleric
Tactical Healer: After an ally has taken damage, been reduced to 0 hit points, or made a Death Saving Throw, as a reaction you can move up to your speed and cast a spell to heal this ally. You can only cast spells from the following list using this reaction: Spare the Dying, Cure Wounds, Sanctuary, Aid, Lesser Restoration, Revivify, Greater Restoration, Heal, Resurrection, and True Resurrection.
Reasoning: I know there are many ways to provide healing and support in 5e to the extent that people compare it to a game a whack-a-mole. But I also have a sense that when a cleric has to use their action to heal, it is not always necessarily their choice but rather an obligation or something they must do because of their role. While it does bring more power to healing potential, it also allows clerics to continue to fight on the front lines while maintaining their position as the party healer.
Druid
Tactical Protection: When an ally within 60' of you is targeted by an attack or spell effect, you can use a reaction sacrifice a spell slot in order to raise protective magic around your ally. Divide the level of the spell slot you sacrificed by two (round up). You add this value to your ally’s AC or Saving Throw Bonus. You can use spells in this way even while you are in Wildshape.
Reasoning: The way I see it, if Clerics are the party healers, Wizards are the artillery, and Sorcerers are the shapers/artists of magic, Druids are the protectors. Their very identity is to protect nature and balance.
Fighter
Tactical Assist: You can strike at enemies to draw their attention, lowering their guard and creating openings for your allies. When an ally makes an attack against an enemy, as a reaction you can move up to your speed and make an attack on that enemy. If the attack is successful, they take no damage. Instead, the next attack to strike this target is made with advantage.
Reasoning: Similar to the Rogue's tactical sneak attack. There are things that make me uncertain about this though, especially in comparison to the Rogue's Tactical Sneak Attack. The Tactical Sneak Attack provides a bonus that can increase as the rogue level's up. On the other hand, the Fighter's Tactical Assist ability can make a critical hit more likely to happen.
Rogue
Tactical Sneak Attack: You can use your sneak attack to make an enemy easier to hit for your allies. During your BFA (Battlefield Action), you can move up to you speed and make an attack on an enemy. If the attack is successful, they take no damage, but are crippled in a way that makes them easier to hit. The next attack made against the target gains a bonus equal to half your sneak attack dice (rounded up).
Reasoning: Allows sneak attack to be used in a new and unique way to assist the party. I feel like this would be the gold standard for a class specific TA that does not require a resource expenditure such as those that require the use of spell slots to use.
Sorcerer
Tactical Metamagic: When an ally casts a spell within 60' of you, you can use your sorcery points to alter your ally's spell with metamagic. As a reaction you can spend the sorcery point cost and apply a metamagic effect you know on your ally's spell.
A sorcerer may attempt to apply metamagic effects to non-spells at the DM’S discretion.
Reasoning: Provides more flexibility to sorcerers regarding how they can use their metamagic, which I think is a good thing. My main problem, though, is how it can be applied to non-spellcasters. Additionally, unlike the other abilities listed, this is not available to the sorcerer from level 1.
Wizard
Energy Conduit: When an ally who is within 60' of you makes a successful attack or spell attack, you can use your reaction to use one of your spell slots to make your ally's attack more deadly. For each level of spell slot you sacrificed, your ally can add one additional die of damage of the same damage type to the attack’s damage.
Reasoning: When I think of classic wizards, I think of fireballs and enchanting weapons to be more deadly. So this seemed fitting to me.
Additional Thoughts for other class design
Paladin: I'm thinking they could maybe put their divine smite ability over someone else's attack. Only problem with this, though, is that it is very similar to the Wizard ability.
Artificer, Monk, Mystic, Warlock, Ranger: I have no idea for any of these. Any help anyone could provide would be welcome.