In this text, extending results of A.Nica and M. Neagu, we consider random permutations which can be written as free words in several independent random permutations: we first fix a non trivial word $w$ in letters $g_1,g_1^{-1},..., g_k,g_k^{-1}$, secondly, for all $n$, we introduce a family $s_1(n),..., s_k(n)$ of independent random permutations of $\{1,..., n\}$, and the random permutation $\sigma_n$ we are going to consider is the one obtained by replacing any letter $g_i$ in $w$ by $s_i(n)$. As an example, for $w=g_1g_2g_3g_2^{-1}$, $\sigma_n=s_1(n)s_2(n)s_3(n)s_2(n)^{-1}$. Moreover, we generalize the problem by allowing to restrict the set of possible lengths of the cycles of the $s_i(n)$'s: we fix sets $A_1,..., A_k$ of positive integers and suppose that for all $n$, for all $i$, $s_i(n)$ is uniformly distributed on the set of permutations of $\{1,..., n\}$ which have all their cycle lengths in $A_i$. For all positive integer $l$, we are going to give asymptotics (as $n$ goes to infinity) on the number $N_l(\sigma_n)$ of cycles of length $l$ of $\sigma_n$. We shall even consider the joint distribution of the random vectors $(N_1(\sigma_n),..., N_l(\sigma_n))$. We first prove that the order of $w$ in a certain quotient of the free group with generators $g_1,..., g_k$ has an influence on the asymptotics of the random variables $N_l(n)$ as $n$ goes to infinity. We also prove that in many cases, the distribution of $N_l(n)$ converges to a Poisson law with parameter $1/l$ and that the family of random variables $(N_1(n),..., N_l(\sigma_n))$ is asymptotically independent. We notice the pretty surprising fact that from this point of view, many things happen as if $\sigma_n$ were uniformly distributed on the $n$-th symmetric group.