[RPG] How does a mount from Find Steed act when unmounted

From page 74 of the 5e Basic Rules:

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.

Or in the PHB p.195:

When you move, you can drag or carry the grapplee with you, but your speed is halved, unless the creature is two or more sizes smaller than you.

So yes, you can grapple someone and then drag them. Now in this case the real unanswered rules question is that you are dealing with an "unopposed grapple" - the wizard isn't looking to avoid it, so does the barbarian need to roll to hit or what? I normally allow such a move to succeed if unopposed, unless it's the prototypical "guy is falling down a pit/off a wagon/etc" situation where there's a question of whether the barbarian could easily lay his hands on the wizard in the first place.

Keep in mind they'd both be grappled, and the barbarian possibly encumbered, while doing all this, though you can disengage a grapple at will. But other than that, sounds like you handled it fine. The barbarian could probably have mounted his horse still carrying the wizard along, as he could also lift him (see the push, drag, and lift rules under Strength, you can do all three to the same weight for a given Strength). Actually getting the wizard to mount another horse would probably incur the same "half movement" cost as mounting it himself.

This answer is wrong. Ignore it, unless it makes your games more fun, in which case, ignore all others.

Yes, you can, and with even more versatility than you thought.

Unlike previous editions or some other games, movement is not a type of action in D&D 5e. It is a resource that you use, almost like a currency that you spend. At the beginning of your turn, you have movement to use equal to your speed. Think of this as a deposit into your account. You can use (spend) movement at any time during your turn, in any amount you have left, any number of times. In between using (spending) movement, you can do anything that your character is capable of, including making attacks, using bonus actions, etc. If you don't have any movement left (account balance of 0), you can't move any more. After the end of your turn, any remaining movement is gone (the account is zeroed).

Using the Dash action increases the amount of movement available that turn by an amount equal to your speed. It's as if another deposit has been made. Therefore, you don't use your Dash to move, you just are able to move more because you Dashed. You can do this at the beginning of your turn, after using some of your movement, or after using all of it.

Now, how does this apply to mounts? Your mount gets a deposit equal to its speed at the beginning of your turn, and you get to direct it how, where, and when to spend that movement, as if it were your own. You can also direct the mount to use the Dash action (no action required on your part), which makes an additional deposit into its movement account. As long as the mount's movement account has movement in it, you can direct it to move as you wish, as many times as you wish, doing anything that your character is capable of in between each movement.

Note: this answer has been contradicted by a Crawford tweet since written.