She doesn't fall off like anything, she has 4% attack speed per level which is the highest of any ADC and average base attack speed and damage scaling, her long range and literal safety net mean with the right team comp she can consider forgoing a defensive item in certain situations. Her reliance on auto attacks as her main damage source and passive steroid mean that she has a 100% up time on her late game damage output so she will be doing her maximum damage at all times without relying on active abilities like Graves or Tristana. Her traps and Q are amazing for turret siege situations as well.
If you want to deal with Caitlyn you do have a few options, you can try out scaling her with certain champions, Draven and Vayne can both outdamage her late game due to their massive damage steroids, though Vayne has trouble in lane vs Cait and Draven takes better mechanics than most ADC to manage axes. Sivir is another solid choice as her kit offers quite a bit to help deal with Caitlyns, boomerang blade offers long range harass and decent damage, fleet of foot allows you to kite her effectively and keeps her from range leashing you, ricochet allows you to harass her and keep her from pushing the wave at the same time, you can use spell shield early game to clear traps for free mana and late game can be used to block her ult. On the hunt can allow your team to rush down Caitlyn before or during a team fight and either kill her or taker her out of the fight. A Jinx Leona lane can have a lot of success vs Cait, Jinx outranges her at level 9 with her rockets, can puch vs her with her splash damage, and Cait has no real answer for the amount of CC that this lane provides. Some good ADC/Support combos: Draven/Leona, Graves/Taric, Sivir/Kayle, Galio/Kalista, Rengar/Nidalee, Riven/Orianna, even Yasuo/Janna if you play it right.
You could also consider a lane swap/kill lane to deal with her, sending a safe laner like Karthus while your ADC and support go top can be an option especially if you are playing Vayne and actually out damage her, alternatively you could swap out just the ADC and top laner depending on the matchup, sending a Vayne top vs a renekton while running a leona pantheon bot lane can both allow your ADC to farm pretty easily and keep the enemy ADC behind not only in gold but also levels. You could also consider asking your mid laner to lane swap for a double AP lane with certain supports, Lux/Morgana for example can be brutal for an ADC to deal with while certain mid matchups can be favorable to an ADC, Lucian and Ezreal can both mid quite well. If that isn't an option you could even run a jungle ADC, Twitch is probably the most common but with the changes to hunters machete that added a DOT effect you can now kite the jungle mobs quite effectively with just about any ADC, I would personally recommend jinx as a jungler since she can clear fairly will with rockets early on and her attack speed helps her kite mobs without taking much damage, just don't attempt the gromp on first clear. Also skirmishers saber into devourer on an ADC is amazing. Some other kill lanes that can work: Leona/Jarvan, Leona/Pantheon, Elise/Annie, Darius/Sion. Some safe 1v2 laners that can work: Yorick, Karthus, Shen, Ziggs, Heimerdinger, Xerath, proxy Singed.
In terms of dealing with Caitlyn in lane keep an eye on her passive stacks and be out of range when it is up, depending on the difference in range you may be able to use dashes to either get into range to retaliate after poke or get from your adc range to outside of her range after a successful trade. If you have an early ganking jungler consider calling for a gank as Caitlyn will tend to be extended quite far into the lane but make sure you clear as much of the incoming minion wave as possible before initiating. As a support clear her traps when it is safe to do so, Morgana is great at this due to her shield and she can sustain and counter push with tormented soil and her passive.
There are also a few champions that outrange her at certain parts of the game or with certain abilities active, these are Twitch(with ult), Kog'Maw(with level 3-5 W), Tristana(level 13+), Jinx(With Q on fishbones level 3-5). Varus can also outrange everything Cait has with his Q.