Are there any encryption schemes that enable to permute homomorphically?
One can do bounded-length "composition of permutations on the ciphertext"

by Evaluating Branching Programs on Encrypted Data.

Public key length and ciphertext length both scale linearly with the number of permutations.

Leveled FHE can evaluate arbitrary-length compositions of permutations, since iterated composition of permutations can be carried out in low depth. $\:$ (So, one does not need full FHE.)

I don't know of any less general methods than that for arbitrary-length compositions.