A sorting network is a geodesic path from $12 \cdots n$ to $n \cdots 21$ in the Cayley graph of $S_n$ generated by adjacent transpositions. For a uniformly random sorting network, we establish the existence of a local limit of the process of space-time locations of transpositions in a neighbourhood of $an$ for $a\in[0,1]$ as $n\to\infty$. Here time is scaled by a factor of $1/n$ and space is not scaled. The limit is a swap process $U$ on $\mathbb{Z}$. We show that $U$ is stationary and mixing with respect to the spatial shift and has time-stationary increments. Moreover, the only dependence on $a$ is through time scaling by a factor of $\sqrt{a(1-a)}$. To establish the existence of $U$, we find a local limit for staircase-shaped Young tableaux. These Young tableaux are related to sorting networks through a bijection of Edelman and Greene.