I seem to have misunderstood the mounted combat rules. Reading them again it seems that the mount and rider are treated as separate entities but share initiative (this assumes a controlled mount, not an independent one).
So the question is: Can a mounted character move using the mount's movement, then the mounted character attacks, then the mount uses a dash action to move away?
(Note: I previously assumed that the mounted character would have to use his action to use the dash or dodge feature if the mount.)
Best Answer
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.