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Does the weight of the grapplee (the one being grappled) factor into the speed the grappler can drag them? Or conversely, is a low strength grappler limited by their dragging/listing capacity when forcibly moving their graplee?

My medium-sized character has 8 strength, so the maximum weight she can carry is \$8\times 15 = 160lbs\$, and her drag weight limit is \$8\times 30 = 240lbs\$. She weighs (plus equipment) over 240lbs as well, so I can also ask about a different character trying to grapple her with only just enough strength to drag/push/list but not outright carry her (~less than 16 strength).

The relevant rules for Lifting and Carrying:

Carrying Capacity. Your carrying capacity is your Strength score multiplied by 15. This is the weight (in pounds) that you can carry, which is high enough that most characters don't usually have to worry about it.

Push, Drag, or Lift. You can push, drag, or lift a weight in pounds up to twice your carrying capacity (or 30 times your Strength score). While pushing or dragging weight in excess of your carrying capacity, your speed drops to 5 feet.

See also the variant rules for Encumbrance, which I am not playing with currently, but still worth considering.

And under Grappling:

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.

So is dragging in a grapple the same as the dragging mentioned in carrying capacity rules? That is do you need to factor the halving of speed before or after (Or at all?) setting it to 5ft, and with the variant rules before or after reducing it by 10ft/20ft when applicable?

I ask, because narratively I don't tend to think of creatures that are 'dragged' in a grapple as being dragged across the floor (which is what I assume the carrying capacity rules mean), more being sumo wrestlers force-walking each other around a ring.

Carrying Capacity vs. Grapple dragging:

As a visual aid, this is the difference I'm envisioning, which is why I don't think this is so cut an dry.

dog with paws splayed out pulled along with a leash Sumo wrestlers grappling, one of whom forces the other back and out of the ring

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