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In the PHB (p 198) there are rules given for mounts:
While you're mounted, you have two options. You can either control the mount or allow it to act independently. Intelligent creatures, such as dragons, act independently.
You can control a mount only if it has been trained to accept a rider. Domesticated horses, donkeys, and similar creatures are assumed to have such training. The initiative o f a controlled mount changes to match yours when you mount it. 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.
However, the rules for animal companions (PHB93) state:
The beast obeys your commands as best as it can. It takes its turn on your initiative, though it doesn’t take an action unless you command it to. On your turn, you can verbally command the beast where to move (no action required by you). You can use your action to verbally command it to take the Attack, Dash, Disengage, Dodge, or Help action.
Nothing in the mount rules say that a mount can take an action when it would be unable to without a rider - only that its actions are restricted to those three.
Can the mount dash, disengage, or dodge while the rider also takes an action?