According to the Barrington's theorem, any circuit in NC1 can be converted to a branching program, whose main operation is the composition of permutations (along with the choosing of permutations according to inputs, which we omit here) . So I wonder whether there is a scheme that can directly encrypt the two-line notation of permutation, which also enables to do the composition of permutations on the ciphertext, except the general methods by using FHE/Multilinear mapping on permutation matrices?
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.