We use the alternate encumberance rules, where you only can carry your Strength times 5 pounds unencumbered. Per the rules, up to that weight, you can move your speed in a round. When you are encumbered you get slowed down by 10 feet, and when heavily encumbered by 20 feet. If the load is so heavy you only can drag it, your speed gets reduced to 5 feet.
What happens if a character has used some movement, and at that point picks up something heavy that alters their encumberance and speed, especially if it is so heavy they can only drag it?
For a practical example from todays game, a character went down in combat. His fighter friend moved towards him (using up 10 feet of his 30 feet of speed). He already was loaded close to the maximum of his unecumbered carrying capacity of 80 pounds, due to armor, weapons, and heavy sacks of loot. The fallen character with their equipment weighed close to 240 pounds. This meant their own equipment, and the fallen character together, were a load of close to 320 pounds. At that point, with Strength 16, that was heavier than the limit of 16 x 15 = 240 pounds for heavy encumberance, so their only option was to drag them -- that works up to 16 x 30 = 480 pounds.
So, can the fighter still drag his friend? The movement gets replaced then by 5 feet of movement, but not all the movement is available to be replaced any more, as a third of it has already been used up.