To be honest Caitlyn is THAT champion I hate midding against, the one and only. Surely there are certain champions I can use to defeat her, but on most, its just not the case. And I think that in probably every single case, she ends up pushing too hard for me (unless shes not very good) and if I want to keep up I have to burn my mana, after which she definitely gets to my turret and starts destroying it. Usually its alot easier with smart play on a caster, as you can simply harass her with your skills, as long as you can harass her harder than she will hit you when you run back to safety, youll be fine. On ranged AD carries, I have a much harder time, if I try to harass her, she will be the one giving the first shot and if I dont, she will have no need to B and simply destroy my turret eventually. Her stats are also higher than most ranged carries, so with trading blows, you will tend to receive more damage, after which point youll try to run, and cait will simply keep shooting at you, especially because you cant really run from her, shes slightly faster than ranged carries like ashe, corki, tristana.
All that being said, you need more staying power than her during laning phase, tons more. Most caitlyns will start with a dorans blade, knowing this, you can take a huge advantage by getting regrowth pendant or rejuv bead(dont forget your hp pots), since youre regeneration will be far greater, even if she harasses you harder, you will eventually push her out of the lane. When you do that, take my advice and b yourself, push lane quick and then b, it is better to b when she does than when you make a mistake and have to b just so you dont get ulted. Chosing when to b is important, dont make her make you b when you cant afford to.
EDIT: I dont really have any tips on ashe other than trying to hit some volleys and landing some autoattacks with your regrowth pendant, other champions simply have better ways of harassment, volley is too easily avoided.