As Eric noted, with old tires, they will overheat quicker and drop off more over a run.
If the kart is handling well but just getting slower, you’re probably best off playing with tire pressures and dropping them down a bit to get the kart to stay in longer.
Never tune a kart on old worn out tires. It’ll react differently to changes than it will on new tires. Often you can chase and chase and chase setup on old tires and you’ll only end up so far out of the tuning window that when you throw new tires on, you don’t pick up any time and you’ve given yourself a handful of new handling issues.