Best special on mounted Eirika

Both are good. But IMO go with Moonbow cause She does'nt have really high of a attack stat to use glimmer to best use so moonbow is a better choice for overall performance but Glimmer is not a bad choice and is a better skill in some situations.

Glimmer is better if you are doing a ton of damage (as in bladetome level damage), or your opponents have generally weak defenses, whereas Moonbow is better if you do less damage or your opponents have high defenses.
Seeing as you’ll be facing swords a lot more than axes, I’d go with Moonbow. Her attack isn’t too overwhelming either.

I actually run Vengeance sometimes on mine, but that's because she's -Res and will get lots of value out of it. Since she'll always be dealing lots of damage thanks to Swift Sparrow, the Atk boon, and Gleipnir (55 on attack, and doubling almost everything), I would go with Glimmer.

Glimmer 100% I have the same Eirika - Swift Sparrow and Gleipner + calvary buffs boosts her attack and speed like crazy. I tried moonbow, but she's far more deadly with glimmer. Also, moonbow is very underwhelming imo.

If you are running Gleipnir, go with Moonbow or Luna. If you are running a Blade tome, go with Glimmer.