Consider $K\geq2$ independent copies of the random walk on the symmetric 
group $S_N$ starting from the identity and generated by the products of either 
independent uniform transpositions or  independent uniform
successive transpositions.  At any time $n\in\NN$, let $G_n$ be the 
subgroup of $S_N$ generated by the $K$ positions of the chains.
In the uniform transposition model, we prove that there is a cut-off 
phenomenon at time $N\ln(N)/(2K)$ for the non-existence of fixed point of $G_n$ and for the transitivity of 
$G_n$, thus showing that these properties occur before the chains have reach equilibrium.
In the uniform successive transposition model, a transition for the 
non-existence of fixed point of $G_n$ appears at time of order $N^{1+\frac 2K}$ (at least for $K\geq3$), but there is no cut-off 
phenomenon.
However, we recover a cut-off phenomenon for the non-existence of fixed 
point at a time proportional to $N$ by allowing the number $K$ to be proportional to $\ln(N)$.

The main tools of the proofs are spectral analysis and coupling 
techniques. Simulations are provided.