Both would need to make saves, or move at 1/4th Speed per turn, but a creature being forced into the area might not have a choice.
Speed is the statistic from which your total movement is calculated from. No distinction on the mode/method of movement is made. The information on caltrops and bearings follows (emphasis mine,)
As an action, you can spill these tiny metal balls from their pouch to cover a level, square area that is 10 feet on a side. A creature moving across the covered area must succeed on a DC 10 Dexterity saving throw or fall prone.
As an action, you can spread a single bag of caltrops to cover a 5-foot-square area. Any creature that enters the area must succeed on a DC 15 Dexterity saving throw or stop moving and take 1 piercing damage.
A creature moving through the area at half speed doesn’t need to make the save.
While grappled/grappling, your Speed is halved, as you are pulling/dragging the other creature, provided you're physically capable of this forced movement.
Moving at half speed does not reduce your speed, and behooves many skill checks that require visual or auditory queues (such as stealth, perception, investigation.)
Average movement speed for most humanoids is 30ft. Half that from grapple is 15ft, you are burdened. Moving at 50% your current speed leaves you at 7.5ft (5ft if using the common 1" grid movement.) One option to get around this is to apply an effect that adds speed (not movement) to one or both creatures.
Applying the speed bonus to one creature, in order to force the other to make the save is certainly an interesting, albeit sadistic tactic.
Eldritch Blast (Repelling Blast) could also trigger the save, provided you can force movement greater than half the creature's speed, and your DM allows it.
Again, no specifications, exceptions, or distinctions on how the creature moves or enters the area of caltrops or bearing is made, unlike things such as
Opportunity Attacks, which specifies how the movement must be made in order to trigger the reaction.
You can make an opportunity Attack when a Hostile creature that you can see moves out of your reach.
You also don’t provoke an opportunity Attack when you Teleport or when someone or something moves you without using your Movement, action, or Reaction.
The wording on caltrops is interesting, as it implies that a creature hovering or flying through the area is also effected