Drag OR Carry
Do weight rules apply?
I am on-record as saying that the Grappling Rules are for creatures, and the Lifting and Carrying rules are for objects. I don't think the weight of the graplee should be considered when adjudicating a grapple, just its size. However, I would point out that the specific game terms drag and carry are not explained in the Grappling rules, but rather in the Lifting and Carrying Rules. There, we find that one can drag or lift weights that are twice what one can carry. This is enough to make me question whether it is as easy for a grappler to carry an opponent as it is for them to drag the same opponent - or in this case, whether there are opponents that they can drag, but not carry, albeit based on size, not weight.
Personal RW Experience
As a DM, I find that the players at my table appreciate it when I interpret rules in such a way that supports verisimilitude in the game. The martial art I practice involves a lot of grappling. I routinely drag creatures around which are larger than I am. On occasion, I even lift them, typically for a hip throw. However, I don't carry creatures that are larger than I am.
The grappling rules state:
Moving a Grappled Creature. When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
Let's focus on the 'or' in 'drag or carry'.
The grammar of CAN / OR
The combination of 'can' with 'or' may indicate a completely free choice, but one in which you are not permitted both:
When you reach the T-intersection, you can turn left or right
(but you can't simultaneously turn left and right).
However, the combination of 'can' with 'or' may also indicate a situation in which you will not necessarily have a free choice, but the possible outcomes are unknown because the initial conditions have not yet been determined:
Mrs. White is allergic to shellfish and might be attending dinner.
Miss Scarlett is allergic to peanuts and might be attending dinner, but will not attend dinner if Mrs. White does.
We have only one sauce dish.
Depending on who attends, we will serve either the shrimp sauce, or the peanut sauce, but not both.
When the guests arrive, we can serve the shrimp sauce or the peanut sauce.
Here we do not have a free choice, but unknown future conditions which force us to use 'or'.
We can serve the shrimp sauce or the peanut sauce.
You can drag or carry the grappled creature with you.
Do I think the grappling rules actually intended the second grammatical interpretation? I honestly have no idea, and certainly can't prove that they did. However, taking them as if they did is one valid interpretation of RAW, dovetails nicely with the use of drag and carry given in the rules, and adds verisimilitude. I believe ruling this way would improve the playing experience.
Moving a Grappled Creature.
When you move, you can drag or carry the grappled creature with you, subject to its size:
When grappling a creature that is a size larger than you, you may only drag it.
When grappling a creature that is your size, you may drag or carry it (your choice).
When grappling a creature that is one size smaller, you may drag or carry it (your choice).
When grappling a creature that is 2+ sizes smaller than you, you may only carry it.
Flapping its wings mightily, your Quetzalcoatlus may drag the brontosaurus across the field, but it may not take it into the air and carry it away.