When you move a grappled opponent, your speed is halved. However, while mounted, your mount is moving you. Is your mount's speed halved when you move a grappled opponent ?
It would reduce only the grappling character's speed, as the other answers explain it, but when taking it strictly RAW you can only use the grappling creatures movement to move the grappled creature: (emphasis mine)
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.
Basically, your steeds movement would end the grappled condition, because it would remove the grappled creature out of the grapplers reach.
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.
The mount's speed is unaffected, unless the weight of the rider and the grappled creature together exceed its carrying capacity (which is unlikely). The main effect is that since mounting and dismounting use half the rider's movement, and the rider's movement is halved when moving a grappled creature, then if the rider wishes to mount or dismount the steed while maintaining the grapple, that will consume his entire movement.
If a character is moving a grappled opponent, that character's speed is halved.
Even while mounted though the speed of the character and mount are independent. Assuming a controlled mount, it just means that the character can direct the mount to act using the mount's actions/speed.
It moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge. A controlled mount can move and act even on the turn that you mount it.
And seeing that mounting a steed while grappling (although very difficult to imagine and probably harder to execute in game) does not in any way mean the steed is now grappling the opponent instead.
Thus, the mount's steed would remain unchanged.
Actually this begs the question of, "Can you maintain a grapple and mount your mount?". Nowhere in RAW, does it say that is possible...but, if I were a DM, I might rule this as possible -- maybe subject to a skill/ability check.
E.g. Something like: Roll to see if you can maintain the grapple this round (If "yes", then continue with the action sequence; if "no", the grapple is lost as per normal rules) and then Roll to see if you can mount your mount (If "yes", then you're mounted; if "no", then you can't mount this round.) Or, maybe reverse the checks, depending on whether you might want to penalize a PC for failing to mount. (I.e. Because you failed to mount, you suffer a penalty to maintaining your grapple attempt)
Now, if we assume that that basic question is, "Yes", then I would probably say the mount would slowed to half speed -- so, that you could maintain the grapple.
While, the other answers seem to think that this wouldn't be the case (slowing your mount), and the movement that's slowed is yours, I would humbly beg to disagree. I would think that the total movement being used should be slowed down -- whether that's yours (unmounted) or the mounts (mounted) -- plus, if any encumbrance limitations apply, add these too. The "in game" rational being, this speed loss is to allow you to maintain the grapple.