Hell, as ANY assassin? I can't just swap targets unlike a Kayle ult or a Zhonyas. If I don't burn anything, Kindred won't use her ult. If I burn everything, then she uses her ult. I can't assassinate ANYONE during teamfights either.
Tips?
The positioning is key. If you can be in the ult, then you can't die either. Then you wait it out saving an ability to ensure the kill. Damage done. Mission accomplished.
If you're Lee Sin, Q her, ult her out of it, and then press Q again
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I can't just swap targets
Why not?
If you are katarina, just cry
Pick this champion lol.... When kindred ults, swap places hahaha.....
Well, you have to scare her into using it. Then, stand on her and just AA her (remember, while inside her ult, you will not die either) and then just throw some kind of burst as soon as it ends. If you time it properly, she'll die. If it's a teamfight, just bait her into using it then run like hell. When the ult wears off, go back in and do whatever it was you wanted to do in the first place -- or, if your team is there, have them CC a specific person / carry out of the circle, nuke them, then finish the rest of their team off when the ult ends.
Stand in the circle with her and wait.
impossible counterplay But huge chance of outplay
Katarina seems to be perfect against Kindred. Ambush, wait for her ult to go off (all in the circle will be low) jump in with combo and do a pentakill. This is only the theory...
It has a massive cooldown. Attack her when she can't use it. Otherwise, wait in the field in the case that you should die for your escape/abilities to come back up and try again. It's not like she'll be full life after the field heals her.
Assassin 
just Q onto them and ham on them until they use their ult then just kick them out of the circle with your ult and kill them.
Ask Singed to throw her out.
Alright, so I'm seeing either wait or run like hell, but in a teamfight, I can get CC'd near the end of the ult so I can't do anything, or if I even step out of the circle, I'll die unless I've got an effective escape like a Talon ult. The support usually has enough CC to peel off of me, so they can reach the ADC in time for the duration of Kindred's ult. Should the frontline be peeling for me while I try to escape to them?
I don't really main assassins by the way. Infact, I'm the worst in them out of all the roles.
Burn most of your things.
Honestly, most people can just auto attack kindred and make him panic then ult, leaving that one skill that damages faster than an auto, especially in the case of LeBlanc, who just has to combo as if her ulti was on cooldown.
U could
Wait for nerfs kek
You don't, to be honest.
The only assassin I've used successfully against Kindred was Wukong, because I killed her so quickly she didn't have time to get out of my knockup before dying. And that only worked because I softened her a bit beforehand. In a team fight it's harder.
Kill Kindred faster than they can react with ult. Rengar or Talon should do the job. Other than that, you can use Zhonyas as AP assassin so Kindred can't use her ult to damage you while being invulnerable themselves. If you are playing assassin with stealth, you can try and use it, and then just wait until it ends, so you don't get damaged (may not always work cause pinks and red trinket, but worth trying). Don't waste your cooldowns, wait for ult to end and then combo target to death. Well, these are some of my ideas, maybe some of these will work.
Super depends on the assassin you're trying to play, since their playstyles can differ.
