Simply damage the rider.

Stay in Saddle: You can react instantly to try to avoid falling when your mount rears or bolts unexpectedly or when you take damage. This usage does not take an action.

It is only a DC5 Ride Check but you could add situational modifiers which might be applicable. I'm a little surprised that there isn't the same mechanic as spellcasting in combat has, where when the spellcaster takes damage the DC is raised by the amount of damage taken. Personally I would houserule that to be the same for riders so taking 12 damage would need a Ride DC17 check to stay mounted.

RAW the rider needs to pass a DC5 Ride skill check to stay mounted when they take any damage.

Simply damage the rider.

The Ride skill says the rider needs to make a check when they take damage under the Stay In Saddle special action.

Stay in Saddle: You can react instantly to try to avoid falling when your mount rears or bolts unexpectedly or when you take damage. This usage does not take an action.

It is only a DC5 Ride Check but you could add situational modifiers which might be applicable. I'm a little surprised that there isn't the same mechanic as spellcasting in combat has, where when the spellcaster takes damage the DC is raised by the amount of damage taken. Personally I would houserule that to be the same for riders so taking 12 damage would need a Ride DC17 check to stay mounted.

RAW the rider needs to pass a DC5 Ride skill check to stay mounted when they take any damage.