When I got my pigs, I set up their cage with a lid. After a while, I expanded the cage and the lid didn't quite work for it anymore, so I took a chance and just took it off. By that time (several months or a year after the pigs arrived), the cats were fine with the pigs. The novelty of scurrying little prey had worn off, and while the removal of the lid did peak their interest, it waned again shortly after. I think they realized that these creatures aren't quite like moths and mice. These creatures don't run from a cat jumping into the cage. No, these creatures COME. AT. YOU. Granted, it's out of curiosity, but the cat doesn't know that. She thinks the pig is coming to eat her. Or something. So she usually turns around and jumps back out, with a look of dismay and disgust on her face.
Now, when we got Marcy, it was a different story. Marcy came to us as a kitten when the pigs were already full grown, so they were bigger than her. She got used to them, they got used to her. She decided right away that fleece makes an excellent bed, and you can still find her curled up in a corner, or stretched out. The pigs are used to her, too. When she jumps in, they come to check out what made the thump into their home. Marcy doesn't care when Sasha (it's always Sasha, Stella is too shy to come out from Sasha's shadow) starts sniffing at her. I even have video of Marcy trying to sleep and Sasha's nibbling at her fur. Marcy is still young, though (she had her first birthday last week) and she plays rough (with fellow cats and her beloved humans). She does sometimes go overboard, get a little big for her britches, and then someone has to step in and remove her from the cage. She doesn't hunt them, exactly. She just sometimes gets into "the zone" and decides to stalk one, try pawing at her. Said pig will spook and book it across the cage and hide under a house, and that's it. If Marcy decides to press the issue, she's gone. These piggies aren't exactly quiet when they feel an injustice has been done to them. If they get touched by a paw, they're wheeking and squeaking (like pigs!) which always gets our attention. But as soon as Marcy comes back to the cage, they're all sniffing each other again, past squabbles forgotten. Definitely something to keep an eye on, of course.
If any other cat jumps in (I have three others, though the skittish one never comes to this area of the house), the pigs know instantly that it's not Marcy, and they turn and run and won't come near the cat. Which is fine by the cat, by the way. Kitty just wanted to jump into the cage so she could jump onto the window above.