Here's a completely untested, basically stream of consciousness idea of how to set up a theatre of the mind system that still uses D&D's stats and core system.
The battlefield is divided into zones, a la FATE. Each zone represents a sizeable portion of a battlefield, and most skirmishes take place on battlefields with at least three (and potentially many more) zones. Under normal circumstances crossing from one zone to an adjacent one simply takes a move action.
Within each zone, you may move around, attack, and so on freely. If you have allies in your zone, you may target them with beneficial effects and include them in your auras, etc. If there are enemies in the same zone as you, you are either engaged or disengaged with them.
If you are engaged with an enemy you are in melee combat with them. They can attack you, you can attack them.
If somebody is in your zone but the two of you aren't engaged, then you are disengaged. You can't attack them in melee combat (yet). You can target each other with ranged attacks, including thrown weapons and short range weapons like darts and slings (though penalties may apply if somebody else is engaged with you at the same time).
You engage with someone by spending a move action to do so. If you try to engage a new target while already engaged elsewhere, you provoke opportunity attacks. You can spend a move action to leave your current melee combat, becoming disengaged from everyone.
Thus anyone in your zone is only ever a move action away, although you risk serious injury if you ignore the enemies in front of you to chase off after another one.
You may only use long range attacks (bows, crossbows, many spells) to target enemies in another zone. Each zone further away the enemy is, you suffer the penalty for a range increment. If you want to close with an enemy in a distant zone, you need only cross into it from your current location, which takes a move action each time you want to leave your current zone to enter an adjacent one.
In zones with rough terrain, you must spend an additional move action to become engaged or cross a zone boundary. Some zone boundaries might also require ability checks or other extra steps to successfully cross. Failing such checks might incur special consequences.
When you cross into a new zone, you aren't engaged with anyone. You may move to engage a new target in this new zone as normal, and following the rules for engagement and disengagement above.
Finally, pushes and other movement effects are handled a little more abstractly. Any push is able to force the enemy it targets out of engagement with one character. For instance, if you have a melee attack that pushes someone X feet, any enemy you hit with it is forced to the "disengaged" state with you. It will take them another move action the next turn to move back to engage you.
You can also use forced movement to push an enemy into another zone.Typically this means that you roll an opposed Strength check, though other abilities may be more appropriate depending on the particular circumstances. The person initiating the forced movement gains a bonus on this opposed check equal to the number of feet they are pushing the opponent. The defender may gain advantage if the situation calls for it, such as their having exceptional stability (a behir with its many legs) or the zone boundary being particularly strong (trying to push an enemy uphill into a tangle of brambles).
Whew! That was a lot more typing than I thought it would be. I wonder if it actually makes any sense! Though now that I wrote it (stealing very generously from FATE, of course) I'm curious to give it a try myself.
