Theory
One of the biggest reasons that D&D combat slows down is that spellcasters have too many spells, and those spells are too complex. It's not easy to decide which one to cast, and it often requires reading long paragraphs of text to determine exactly how the spell works.
Fighters don't have this problem, as they have fewer options, and those options are simpler. At worst, this is a frustrating balance problem, as they pale in comparison to full casters; at best, it is a call to action, which drives fighter PCs to be more creative with their actions, and with a good DM, this can both level the playing field and enhance gameplay.
4e already tried turning fighters into wizards. It didn't work. But what if we could turn casters into fighters? What if, instead of having hundreds of spells, casters had just a few core powers, but those powers were very flexible. Consider powers like Wild Shape, or Channel Divinity, or warlock invocations; those not only achieve what's described, but are more flavorful and achieve better class fantasy.
Here, then, is an attempt to describe such a system, that fits within the bounds of 5e.
Implementation
A Power is any sort of supernatural ability possessed by a creature. Many monsters have powers due to their nature, and some PCs have powerful races that supply powers. But the bulk of powers practiced by PCs are likely gained through class abilities, specifically from spellcasting classes.
Generally, a class will grant access to a list of powers, and grant knowledge from that list according to a schedule. Skill Alternatives are often available to gain even more power knowledge. For example, a given class might offer 2 Mana per level, 2 powers known at level 1, and an additional new power every 5 levels.
Powers are generally very broad in scope, typically describing what is happening as opposed to the effect. Thus, while a 5e spell describes an effect, such as "a 20-ft cone causes 4d8 fire damage", a power describes a cause, such as "you have the power to create bursts of flame", followed by a list of possible ways it could be used, such as forming it into a cone, narrowing it into a line, making it stick to enemies, etc. This list is not meant to be exhaustive, but more a set of guidelines, much like the way Skills work.