Is your cat scratching his paws until they bleed, or just licking them? A move can definitely cause stress, but bald patches on his belly - that's already a serious signal that needs a vet's attention. Stress usually shows up as anxiety and increased licking, but it doesn't typically lead to such pronounced hair loss.
The problem could be stress, allergies, parasites, or a fungal infection - without an examination from a specialist, it's hard to say. The key thing is that his hair is falling out and visible bald spots are appearing - that's reason not to put off a vet visit. If your cat is wounding himself to the point of bleeding or the patches are spreading, go urgently. In the meantime, you can try to reduce his stress: keep familiar scents around, create hiding spots, don't bother him unnecessarily - but that won't replace a proper diagnosis.