The target of a grapple will be held aloft by the grappler, both creatures will fall, or grappling between two airborne creatures is impossible
Introducing falling to the scenario is an interesting wrinkle, but it shouldn't change much about the underlying situation. The operative elements seem to me to be (1) what does it mean to grapple, (2) what does it mean to carry, and (3) what does it mean to lift? There is then a secondary question of whether or not falling is an effect which would force the grappled creature away from the grappler in the same manner as Thunderwave.
If you want to stick with RAW, the direct application of these rules leads to one of the conclusions in the header: the grappler lifts the target and both stay aloft, the grappler does not lift the target but the grapple continues and both creatures fall together, or this sort of airborne grappling is meaningless.
1. Does the grappler maintain their grip?
On solid ground this isn't very controversial. The grappler gets a hold on their target and imposes the Grappled condition. This prevents the target from moving away (because their movement speed is 0), representing that the grappler is holding the target in such a way that they can't move away.
There is no way to disambiguate imposing a grapple (described as seizing the target in the rules) from having a hold on the target; on the ground, the target's move speed of 0 means that the grappler prevents the target's movement and so they are held stationary (within a grid square, at least). The grappler prevents the target from moving (in the case of something like a wrestling pin) or resists their efforts to move (like pulling target back every time they try to move).
This should work the same way in the air, but with the extra effect that the target of the grapple cannot remain in the air on their own due to the movement speed reduction. And so here we reach the first consequential decision: either the grappler maintains their hold and the target keeps the Grappled condition or the grapple ends. So let's look at that:
1a. The grapple ends
This is an interesting case and hinges on whether or not falling is considered an effect that would force the target away from the grappler, like the Thunderwave example given in the condition description for Grappled. I don't see any compelling reason in the rules to accept this interpretation, but neither do I know of an explicit entry which contradicts it.
If the ruling is that falling is an effect that causes the target to be removed from the grapple then they are immediately freed from the grapple, restoring their movement speed, and so the grapple is wasted and fundamentally self-defeating.
1b. The grapple does not end
If the ruling is that falling is not sufficient to immediately end a grapple then the grapple is maintained even though the target's movement speed is 0. That means one of two outcomes are possible: the grappler stays the same height above the ground and holds the target aloft with them, or both creatures fall together. The latter case is easy, in that you would use the falling rules in the PHB or Xanathar's Guide to Everything. The former case requires a judgment call.
On the ground, the ground itself does the work of keeping the creatures from falling. But in the air that won't apply, and so if nothing keeps the target in the air they will fall. In the most general case that's going to be the grappler's own ability to keep the target aloft. As far as I can find the rules do not cover this situation at all, and so the strictest possible interpretation suggests to me that both creatures would fall together.
It's not unreasonable to suggest that the grappler keeps the target in the air. For ground-grappling it is mechanically irrelevant to the Grappled condition if the grappler pins the target to the ground or lifts the target's feet off of the ground. If the grappler has a firm enough hold on the target to impose the Grappled condition in the air there are no rules indicating that the grappler can't use their own strength to prevent the target from falling. How that would be adjudicated is separate.
2. What does it mean to grapple, carry, or lift?
Grappling a target requires using at least one free hand to attempt to seize a target. If successful, the target's movement speed drops to 0. The grappler doesn't need to carry the target to accomplish this, and so the grapple itself is enough to prevent any efforts on the target's part to move. Carrying only applies when the grappler wants to forcibly move the target to another space on the map-- the grappler does not need to engage in carrying to keep the target stationary, as the Grappled condition already imposes that.
If you're not going with the (1a) interpretation above, then the grapple continues and we have two more choices on how to adjudicate the grapple:
2a. It doesn't cost anything to keep the target stationary, even in the air
This is how grappling works when at least one creature is on the ground. It's not realistic, of course, to rule that the grappler just automatically holds the target in the air with no other considerations, but that's where "D&D isn't a physics simulator" comes up. The size class restriction on possible grappling targets may account for this, as it does on the ground. It's a bit weird, but seems rules-complaint to me.
2b. It takes effort to keep a target stationary when they and the grappler are in the air
Holding something above the ground despite gravity is equivalent to lifting that thing, and so you could apply the grappler's Strength. Because the target's speed is 0 the only way for it to stay aloft is for the grappler to hold them. The most direct way to do that is to check the grappler's lifting or carrying capacity against the things they're carrying plus the weight of the grapple target. Creature weights are generally not defined, so that isn't going to be very precise. It could also be represented by a skill check, though that will be similarly imprecise.
If you follow this interpretation, then the grappler will have at least the potential to keep the target aloft despite its zero speed. If the grappler doesn't do that, either through insufficient lifting/carrying capacity, a failed check, or a lack of interest, then they and the target will plummet together.