There's no way you could safely catch those cats by hand. And if they are careful and patient in trying to get them in the boxes, they'll probably get loose and need to be caught again. You gotta get them in the box NOW! Sadly, when they go in situations like this, most of the cats end up euthanized. They are either so sick that it's just the most kind thing, or they are so incredibly wild that they won't make adoptable pets. I saw one episode where a good many of the cats ended up at a cat sanctuary, though. They do what they can. One episode had like over 140 cats in a place. The workers had to wear respirators and protective suits to protect them from the ammonia and feces, and it took them two days to get the cats out. Most of those cats were euthanized due to severe repiratory problems from living in such filth.
I wish we had stiffer penalties for that sort of thing. I understand that they can get out of hand if not fixed, but there's too many low-cost spay and neuter clinics around now for that to be an issue. Especially in a place like Detroit or NYC when all you have to do is call the SPCA and tell them "Look, My cat had a litter and now they are getting out of hand, I need some help with getting them fixed." One of those huge seizure efforts was due to ONE female cat having a litter of kittens. They didn't get them fixed, so they were inbreeding and just producing more and more cats. The guy said he was spending like $1400 a month on cat food and his house was so nasty it was probably condemned. And most of the cats were euthanized. There's no excuse for it. There was at least 6 weeks to two months between the first and second litters when he should have done something about it, but didn't. No excuse.