Slitherine

How exactly does the formula for the pursuit look.
I know that flanking modifier affects the pursuit but I don't know exactly what. Now I can't make the decision whether it's better to buy a regional unit instead of a regular or light infantry because I don't know how much more they inflict losses during the pursuit.

During battle flanking is a bonus to attack rating when rolling for the losses the flanked will receive. During pursuit, flanking almost work the same, this is a bonus to attack rating of each winning unit. If you beat a 10-sided die, then you inflict 1 damage to the enemy. You then remove 5 to this modified rating, and then you roll another die, if you succeed again, again +1 damage. You then remove 10 (total) and roll a third die ... etc.

So basically, each flanking value is 10% chance to increase damages in 2 circumstances.

Does the pursuit work in the way that every single unit you have in the army has a change to deal damage?

No there is a limit, it depends of the number of duels you won in the melee phase, and there is a bonus for each flanking you did. But having a lot of good flankers in your army will make the pursuit damages higher in the end.