It's kind of my greatest conundrum right now. For one thing, I think I need to use an animation with root motion to properly position the enemy during the launch - I'm doing it with phsyics right now, and the result is very inconsistent- sometimes they go too high, usually they go too low, and sometimes they just get stuck underneath the player - that gif was a rare case of things just working nicely
The best feeling input method is also a puzzle for me. Right now it's in the style of DMC Devil May Cry - with it's own dedicated button. Tap it to launch the enemy but stay put yourself, hold it to fly up with them. The problem here is that this means I'm using up all the face buttons without including a ranged attack yet (X = Light Attack, Y = Heavy Attack, B = Launcher, A = Jump). It also begs the question of what to do when you hit the launch button while you're in the air. The obvious solution would be to make you smash back down, but the heavy attack already does that. I could make the heavy attack do something else though, and that might be a good solution if I can think of something - there's plenty of other buttons I can use for shooting after all.
The DMC1-4 approach always feels fluid, and doesn't require an extra attack button, but High Time in those games requires a lock-on feature, which doesn't fit my vision of the game. To me it just doesn't make sense for the style of camera I'm using.
There's also the God of War approach, where you just hold the triangle button to turn any combo into a launcher. While I like this in theory, it excludes the ability to keep yourself grounded (unless you wanna have two levels of long-press, but that sounds clunky). I always loved combos in DMC that include knocking up the enemy and then juggling them with ranged attacks while you stay on the ground, so I definitely want to keep that in for when I have ranged attacks.
The Ninja Gaiden approach is definitely one I'm thinking of trying though. In ninja gaiden, you have two ways to launch enemies. You can either X -> Y to do a very quick knockup, without the option to stay on the ground (I think, maybe you can hold Y to stay rooted?) or you can tap on the analog stick in a direction and hit the heavy attack button. A short tap will knock up the enemy, while a long press will send you up with them (or maybe it's the other way around?). I typically used the former approach, partially because ninja gaiden's enemies are very dangerous, and less time spent on the ground means less time getting mauled by them, but also because the stick + Y attack had a much bigger windup, so it was very easy to get hit while executing it. If I kept the short window for my current launcher (which does include some i-frames at the moment), and allowed you to use the launcher at any point in a combo (as DMC does), it might work rather well. I'll definitely have to play around with it.
Edit: Realized I didn't include Bayonetta in my breakdown. Mainly because it has the same problem as Ninja Gaiden - the launcher is tied to specific combos, which you have to remember. Combo lists in Bayo are so huge that I never quite was able to memorize them completely- normally I ended up using the launcher by accident as much as on purpose. While I have no problem with approach in these games, I want to follow the DMC approach for my game, because I feel like being able to take the fight airborne at will is bit part of what makes the game feel so free-form.