Inspiration in D&D is like 60% of the way there. I tried to get my old D&D group to use it, but they were too D&D-mindset to really embrace it. By that I mean kind of passive, and very zoomed in on their character rather than the more "writer's room" view Fate favors.

But, if you have a good group, you could stitch it onto D&D 5e without much problem. I'd do it like this:

• Replace inspiration with Fate points.
• Remove the inspiration cap. You can have more than one.
• Begin each session with 3 fate points. (Adjust to taste, but 3 is the default fate points in Fate, so.)
• Fate points can be spent per Fate
  • When making a check, if you don't like the roll, you may spend a fate point for a flat bonus or reroll if you can sell the group on how it ties into your character or the scene. For example, if your character has a pirate background and you're trying to run across a ship's deck in a storm, you could spend a fate point to be like "...sure it looks like I'm going to faceplant, but I've been in rougher seas before as a pirate"
    • I'm unclear on the math but I think a +4 flat bonus would be fine.
  • You may spend a fate point to declare a story detail if you can convince the group this is a good idea. This should tie into an aspect of the scene or your character, but ultimately the table must accept an idea. For example, if your character concept is "The 2nd Best Wizard from Tarant", and you find all the scrolls in town are sold out, you could suggest your nemesis is responsible
• Import Aspects into the game. Read about them in Fate's SRD
  • Require every important character (that's the PCs and key NPCs) to have a High Concept. That's a tag line. Read about it in fate. It might be "Salty Sea Wizard", "Lucarian Royal Assassin", "Intrepid Delivery Girl", or whatever. Don't try to power game this
  • Require every important character to have a Trouble. Read about that in Fate, too. "Sucker for a smile", "Unlucky Gambler", "Memorably Weird Looking", or whatever. These should suggest interesting ways they can come up
  • Encourage players to do the phase trio (see fate srd) to generate more background aspects. For example, they made decide that two characters met when one accidentally summoned a demon in the library, and the other helped banish it. That's a "Will read any book" and a "First name basis with at least one demon" pair of aspects, maybe.
  • Add aspects to scenes. Slap them on an index card, your dry erase board, or your virtual tabletop. These are your "Stacks of unmarked boxes", "Roaring Thunderstorm", "There's blood everywhere oh gods" short phrases that describe part of the scene. You want the players to interact with these. Get your players to suggest some
• Award fate points when players' troubles make interesting things happen. When the "Unlucky gambler" puts his family sword up for ante in the back alley dice game, that's a fate point
• Bribe players with fate points to let interesting things happen. When the player is being boring and doesn't want to explore the old tomb, invoke their "Lover of History" aspect to encourage them.
• Consider letting players gain or improve aspects on level up. You want a player to be able to change from "Junior Wizard" to "Archmage of Elleife's Court" over the course of the game.

Ok. Lots of text. Bear with me.

The big thing that comes with importing all of this is the creative burden on your players explodes. Players need to be thinking about their character's backgrounds, the scene's aspects, and how they can use them for bonuses or compel themselves for fate points. You cannot just phone it in like you can with D&D. Because let's be honest, a lot of D&D players are not being their most creative selves. They slap down "Dave the fighter" and are good to move 30' and attack. If that is the kind of game mode you want to play, have at it, but none of this stuff will work well with it. But if you do have players that are creative, and are willing to engage with the system, this all can really sing.

Most of the stuff there exists at the narrative level, so it would be easy to import into other systems that don't have much stuff operating on that level. That's most close relatives of D&D, but not most PbtA games, probably. If there's already metacurrency, it might clash.

The second thing is that it does provide a power bump to the players. They gain a new resource they can use to bump their rolls. You can address this by slightly increasing the strength of encounters. You can also just let the players be more awesome. You can also spend fate points on your NPCs (the important ones have aspects and troubles, right?). The "Duelist of Legend" pirate captain can spend a fate point to bump his miss up to a hit when sword fighting the PC. (But they couldn't invoke that aspect to run away, because that's not very Duelist of Legend, is it?)

This can partly replace legendary resistance in 5e. Your lich has "Defies Death Itself" as an aspect, so he can spend a fate point to bump his save versus whatever. (You probably want one fate point per player in your GM pool in a scene, but adjust to taste.)

Anyway. I think about this a lot because I was playing D&D weekly for the past ~3 years, and I got really sick of it. As you can see, I'm happy to go on and on about it.