The description of Ashardalon’s Stride spell reports:
3rd level transmutation
Range: Self
Components: Verbal, Somatic
Duration: Concentration, up to 1 minute
The billowing flames of a dragon blast from your feet, granting you explosive speed. For the duration, your speed increases by 20 feet and moving does not provoke opportunity attacks. When you move within 5 feet of a creature or object that isn’t being worn or carried, it takes 1d6 fire damage from your trail of heat. A creature or object can only take this damage once during a turn.
My question is, how does this work when you’re mounted? Does the horse gain the twenty feet of movement, and it just carries you along with it? Do you have to dismount to use it? Do you shoot off your horse because of the explosive force of the spell?
Best Answer
The horse does not gain anything from the spell.
The spell has a range of self: this means that it targets you and only you, no others. Indeed, from the PHB we can read in the Range section (emphasis mine):
Moreover, the spell description clearly states that all the effects apply to you, the caster.
This works if your mount has been created by the Find Steed spell.
The Find Steed spell (or the higher level version Find Greater Steed) allows the caster to conjure a steed, and a nice option provided by such spell is the following one:
In this case, Ashardalon's Stride can target both you and the steed, giving them firing hooves, increasing their speed by 20 ft and all the other cool effects.
A DM could consider the creatures affected by the flames of the stride as targets of the spell too (see the discussion about What counts as a target for a spell?): this interpretation does not allow the Ashardalon's Stride to be a valid spell to be used in conjunction with Find Steed.
If I were the DM I would allow it, even if the above interpretation is correct, because of resources spent (2 spell slots) by the player and because the scene would be really cool.