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The skin surface looks like liquid latex. That is going by the way it folds at the joints, or anywhere it bends. Some of the parts of the puppets look like plasticine, which allows her to re-sculpt and make changes. Some does look like the latex has been painted over the plasticine. And some things, like the pig, look more like the body was either cast in a plaster mould to form a thick skin, or painted onto soft urethane foam sheet to form a skin surface - it folds like there is nothing underneath in places. I see that there were hard "bones" on many of the puppets, probably either epoxy putty or polymer clay like Super Sculpey, so the shape of the limbs was there under the latex skin. It would only go out of shape at the joints. I do that too, but with a little bit of foam in the joint to blob the latex onto. I only use plasticine as a sculpt to make a mould from, and then cast in silicone or foam latex, I don't use it in the finished puppet.
I did try painting latex onto a small plasticine baby dinosaur figure once, and it did let me do a nice paint job. The latex skin stopped me leaving fingerprints, but after a few frames the clay underneath started to break up, and I couldn't smooth it over and blend it back together like you do with claymation. It was just clay with no armature inside, so that was a bad idea and I didn't do that again. For very small puppets like that, and some big ones, I switched to building up with urethane foam, and dabbing on latex, layer after layer to build up muscle shapes, and that gave me a more durable puppet. I didn't have to worry about the clay breaking up underneath the latex skin.
But here the crudeness is part of their character and style, so I would probably go with a build-up method as she did, just not with clay inside.