| Troubleshooter |
It doesn't matter where the zombie is moving TO, it matters where the zombie is moving FROM.
If the zombie is moving from a threatened square, then threatening creatures can make an attack of opportunity against it.
If a zombie doesn't exit a threatened square and ends up adjacent to a rider and his mount, they can't do anything about it.
If a zombie starts out in a threatened square (such as adjacent to the horse) and moves to another adjacent (threatened) square, it provokes.
The typical horse doesn't have any more reach than an armed human, but if rider has a Reach weapon, then the zombie would have to exit a threatened square to reach him. One of the classic perks of mounted combatants.