Quite honestly, I think you're going about this the wrong way. Kalista is made to be a strong kiter, she's supposed to be good at preventing you from catching her. She has a 45% slow on her e if you start powerballing towards her that makes it quite easy for her to stop you from catching her, and she can constantly jump out of your range.
Why would you try to play against a champions strengths? Would you ever try to outburst a leblanc? Would you try to duel a lee sin at level 3 with amumu? Of course not.
Kalista's biggest strength is her ability to avoid being hit by melee champions through kiting, and her amazing ability to avoid skillshots with her passive. Don't play against Kalista's strengths, play against her weaknesses instead.
For example, if she is stuck in the middle of a bunch of people, getting out is quite difficult for her, and she'll usually die. Her hop isn't as long-ranged as corki's e or tristana's w, and she doesn't have vayne's invisibility.
I would say her biggest weakness though is that she's quite weak to other ranged champions. Compared to other ADCs, Kalista doesn't deal as much damage (With some exceptions such as Ashe).
Instead of trying to chase down Kalista, which plays to her strengths, focus on her weaknesses instead: Protect and peel for your ranged damage dealers/the people who can kill Kalista, and try to give them the opening to destroy Kalista.