Flier Team building help

I don't have any of the mages, so I can only talk about the melee version (struggles with B!Ike and other such physical tanks.)
I'm using a pseudo goad stack with Cherche(SP farming), Hinoka (hone), Cordelia(goad) and Elincia(goad).

My Elincia is +Spd, and under Hone reaches an impressive 52/43 offence, allowing her to run DeathB3. A neutral Elincia reaches 52/40 under hone so may prefer DartB3 instead. This all depends on how well you position with goads. In my opinion, Elincia is the pinnacle of Flier Emblem melee, having reasonable bulk and an incredible stat line. Definitely use her.

Having a hone available to boost your units is a must have, even on goad stacking teams. So naturally, Hinoka comes as well.

Green Fliers aren't a neccecity, but having a unit that can handle lances is important (duh). Elincia can sometimes turn the tables on blues when desperate, but Cherche gives some Opponent Phase viability to your team. Additionally, when honed and goaded, she's not slow.

Where I have Cordelia, you can use a flier of your choice! I see your Cordelia is -Spd (same here) so you can use any of your fliers. Subaki and Caeda provide some Opponent Phase bulk (Def and Res, respectively) while Tana edges more to the offense side.

Finally, while the majority of the builds are pretty standard, it is VERY IMPORTANT to have 4 REPOSITIONS! When against Horse units or Archers (or mounted archers...) you are quite outranged. In that case, you can use reposition to fling a Flier at the troublesome mage or Archer that impedes your advance. They take out the archer, and survive the next Phase - depending on the colour, number and weapons of the enemies, consider carefully - so you can fly in and clean up for a victory. Reposition is important. Abuse that unrestricted flying power for a victory! Beware of man-made boxes ;D

Cheers!
-Chub

I ... dont think you have the materials needed for a single, functionnal Flier emblem team.
The arena has two main threats, that often work in a pair : Mr Tiny hands and "THE ENEMY".
And you need to counter AT LEAST 1 of the 2 for your team to be viable, and by "counter", I mean you need to be able to bait the enemy and kill him on the enemy phase, while it doesn't require you to charge into the whole enemy team.
You might want to max the Deflect Magic seal to deal with Reinhardt using Summer Corrin (She is naturally way faster, so unless Reinhardt has Blue tomebreaker, you are safe), and use her along with a brave weapon user to kite threats like Archers. Another important note : Amiti Elincia loses to BIke because of Urvan+Aether.

Your flier team is similar to mine, where I run Cordelia, S!Corrin, Cherche, and Elincia.

My Cordelia (+Spd) runs Firesweep Lance+ with Heavy Blade, Hit & Run, and Galeforce. This basically makes her a safe Brave Lance user that can also apply chip damage to bulky units where it's needed, even if she can't 1RKO them, e.g. Effie, B!Ike, sometimes Black Knight.

I also include fortify fliers for S!Corrin (Blarblade+), which lets her safely bait and kill Reinhardt. I run her with the Close Counter/Vantage set, so she can easily bait and 1RKO reds, most blue mages, and can even 2RKO B!Ike if she is in Vantage range.