This is a slightly foggy area...but thankfully, Jeremy Crawford (official voice of rules for WotC) has weighed in on this in several (indirect) ways...
Q: what are the rules for creatures carrying friendly PCs? e.g mage polymorphed into giant eagle carrying allies? (5e)
A: See "Lifting and Carrying" in the Player's Handbook (p. 176) for rules on carrying capacity.
Additionally, in relation to grapple/drag...
Q: is pushing/dragging a grappled creature subject to the carrying capacity rules?
A: The rule on moving a grappled creature (PH, 195) works regardless of a creature's weight. It cares about creature size.
So, this gives us a baseline to build on top of.
Crawford has said that you can carry a friendly (and thus, willing) target around with you using the Lifting and Carrying rules. This seems rather evident, because horses don't have to grapple you to carry you (or someone you captured and threw over their haunches) around. Additionally, giving someone a piggy-back ride is not at all like seizing someone in combat. Furthermore, he states that the Lift/Carry rules do not apply to Grappling.
From here, we don't have any further clarity from the developers, but we can move on to plain English definitions to clear things up. Because that was the intent of D&D 5E is that plain English definitions could be used to help adjudicate the rules.
Grapple: to seize and struggle with another
Based off this definition, we can work out what 'grappling' is. A "Grapple" is when you have seized someone who does not want to be seized. They are resisting you and trying to get loose.
If someone is not resisting you, you are not grappling with them. If someone is not capable of resisting you, you are not grappling with them. You're just holding them. Throwing an unconscious or willing individual over your shoulder is not grappling, that's carrying.
Thus, the ruling that makes the most sense, based off both Crawford's rulings, simple definitions, and logic--is this:
"If a target is unwilling, you are grappling them. If the target is willing or unable to resist, you are carrying/dragging them."