It doesn't, and as far as I know, the closer you are to it's center the more damage it does.
I take exhaust on... Well, almost everyone.
And you don't really need to play harder CC champions, it just helps to know how the other champion is playing. Fioras, I've noticed either blow their ult right away or do a couple of AAs and then use it. Most rarely alter their attack patterns, so if she is wrecking you, take note of how she's attacking. If she's waiting until she takes some damage, go ahead and hold onto your exhaust for a second, then use it. This should take care of a decent amount of damage.
That, combine with a form of healing or, preferably shields if she's using ignite, will reduce the chances of your ADC dying. If you are Lulu, this is easier in my opinion, because you can use her polymorph or exhaust, and if that fails, go ahead and ult your ADC and shield her with Pix.
You could also forgo the use of flash. I use it, but I can also drop it if need be for an extra summoner like heal. Flash is a GREAT summoner, but if you know how to play and are in the lower leagues of skill (like most) you can work around it. I however, don't recommend this every game, and it will be a judgement call on your part.