Spellcasting from flying mount

[RealAlchemy]

I had the idea of creating a size small wyldblooded sylvan bloodline sorceror, intending to fight using spells from the back of a size medium flying animal companion such as a roc, giant wasp, or pteranodon. How can I improve the concentration check for casting from the back of a moving mount, and what is required to use the spell partway through the mount's movement in the manner of shot on the run? Ride by attack and flyby attack don't seem to apply here.

[Rerednaw]

I believe this covers what you were looking for.

PRD wrote:

Vigorous Motion: If you are riding on a moving mount, taking a bouncy ride in a wagon, on a small boat in rough water, belowdecks in a storm-tossed ship, or simply being jostled in a similar fashion, you must make a concentration check (DC 10 + the level of the spell you're casting) or lose the spell.

As for improving the check, Paizo limited it so casters wouldn't have auto-success in all situations. But there are still a few feats and traits you can take. Such as:

Ultimate Magic wrote:

Uncanny Concentration
You have learned to enter a deeper state when casting spells, allowing you to shrug off distractions, damage, weather effects, and even the effects of other spells.

Prerequisite: Combat Casting.
Benefit: You do not need to make concentration checks when affected by vigorous or violent motion or by violent weather. You gain a +2 bonus on all other concentration checks.

Also riding a mount means doesn't means it takes additional actions, especially if the mount is combat trained.

Hope this helps.

[avr]

On the other part, 'what is required to use the spell partway through the mount's movement', the answer is nothing is required at all besides the concentration check. It's probably not a melee attack which is the only thing restricted, and unless there's a ranged attack roll the mounted archery stuff doesn't apply.