Sentinel does not prevent a creature from gaining new modes of movement or increasing their movement speed after it has been applied.
I'll quote the full text of the Sentinel feat (PHB, p. 169-170), just so we know we aren't missing anything:
You have mastered techniques to take advantage of every drop in any enemy's guard, gaining the following benefits:
When you hit a creature with an opportunity attack, the creature's speed becomes 0 for the rest of the turn.
Creatures provoke opportunity attacks from you even if they take the Disengage action before leaving your reach.
When a creature within 5 feet of you makes an attack against a target other than you (and that target doesn't have this feat), you can use your reaction to make a melee weapon attack against the attacking creature.
The first function of the feat ("speed becomes 0"):
- Whenever you hit a creature with an opportunity attack, its speed drops to 0 for the rest of the turn. This stops any movement they may have been taking.
Let's break that all down, part by part.
When you hit a creature with an opportunity attack,
Here we have the triggering condition. This tells us that this only applies on opportunity attacks, which are typically caused by creatures trying to move away from you. So it doesn't apply on regular attacks from an Attack action, and it doesn't apply on Readied attacks.
the creature's speed becomes 0 for the rest of the turn.
Next, we have the effect that is triggered. A creature's speed becomes 0. It doesn't spend the creature's movement, and it doesn't subtract movement from the creature; it sets its speed directly to 0, and that setting lasts for the rest of the current turn.
Do you know what is missing from this portion of the ability? It does not have any clause stating that it prevents a creature's movement from being increased again after it has been applied. But maybe there's something else in the ability that says otherwise. Let's read on.
The second function:
- Creatures provoke opportunity attacks from you even if they take the Disengage action before leaving your reach.
This doesn't really matter much in this situation, it only provides an additional situation where you can spend you reaction to take an opportunity attack, and has no effect that interacts with the speed reduction mechanic.
And the third function:
- When a creature within 5 feet of you makes an attack against a target other than you (and that target doesn't have this feat), you can use your reaction to make a melee weapon attack against the attacking creature.
This also only provides an additional situation in which you can make an attack as a reaction (though it's not an opportunity attack): when a creature within 5 feet of you attacks someone else who does not have the Sentinel feat. Again, it has no interaction with the speed reduction mechanic.
So where does that leave us? What do we now know?
We know that:
- When you hit a creature with an opportunity attack, the creature's speed becomes 0 (and since it has a speed of 0, it can't continue moving)
- You can make opportunity attacks against a creature even if that creature has taken the Disengage action
- You can make an attack as a reaction if a creature within 5 feet of you attacks someone else who does not have the Sentinel feat.
What do these things tell us? Well, we know that:
- The speed change is not a speed reduction. We don't spend their movement, and we don't subtract anything. The speed is set directly to 0 feet - do not pass go, do not collect $200.
- There is no clause preventing further speed modifications. If we cast a spell, or activate a special ability, or do some other action that changes our movement speed after Sentinel is applied, the Sentinel feat does not have a clause preventing this.
But how do we know that this is the case?
For comparison, let's take a look at the description of the grappled condition:
- A grappled creature’s speed becomes 0, and it can’t benefit from any bonus to its speed.
- The condition ends if the grappler is incapacitated (see the condition).
- The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the Thunderwave spell.
There is a clause in the grappled condition that Sentinel does not have: "it can’t benefit from any bonus to its speed." Without this clause, Sentinel does not prevent the target's speed from being changed after Sentinel's effect is applied.
Some examples of what does and doesn't work:
These do work, but only if used after getting hit by Sentinel:
The spell longstrider has the effect: "The target’s speed increases by 10 feet until the spell ends." 0 feet + 10 feet = 10 feet, so the target now has a speed of 10 feet.
The spell fly has the effect: "The target gains a flying speed of 60 feet for the duration." This has two effects. First, it grants the target the ability to fly via a fly speed; second, it sets that speed to 60 feet. Since it sets it to a specific value, it overrides the Sentinel feat which has also set it to a specific value.
This doesn't work:
The Tabaxi's racial trait Feline Agility (VGtM, p. 115), which states: "When you move on your turn in combat, you can double your speed until the end of the turn." Whether it's used before or after getting hit by an opportunity attack from someone with Sentinel, it doesn't help. If it is used beforehand, then we have your speed doubled to 60 feet but then reduced to 0 feet by Sentinel. If it is instead used afterward, then we are doubling the speed we were set to by Sentinel - in other words, 0, because 0 times anything = 0.
And just a reminder for those who may have forgotten: There are no hidden rules in D&D 5e. Abilities spell out exactly how they work. If this ability were intended to prevent a target from further changing their speed, it would say so.
Similarly, there is a difference between can't move and speed becomes 0, as discussed in this question. Further, in a comparable situation, if a creature's speed becomes 0 due to exhaustion, as asked in this question, nothing prevents it from magically (or otherwise) increasing its speed.