Following Talagrand's concentration results for permutations picked uniformly at random from a symmetric group [Tal95], Luczak and McDiarmid have generalized it to more general groups G of permutations which act suitably 'locally'. Here we extend their results by setting transport-entropy inequalities on these permutations groups. Talagrand and Luczak-Mc-Diarmid concentration properties are consequences of these transport-entropy inequalities. We also consider transport-entropy inequalities on G for a larger class of measures. By projection, we derive transport-entropy inequalities for the uniform law on the slice of the discrete hypercube and more generally for the multinomial law. The transport-entropy inequalities settled in this paper are new examples, in discrete setting, of weak transport-entropy inequalities introduced in [GRST14].