Starting from first principles, we derive the fundamental equations that relate the $n$-point correlation functions in real and redshift space. Our result generalises the so-called `streaming model' to higher-order statistics - the full $n$-point correlation in redshift-space is obtained as an integral of its real-space counterpart times the joint probability density of $n-1$ relative line-of-sight peculiar velocities. Equations for the connected $n$-point correlation functions are obtained by recursively applying the generalised streaming model for decreasing $n$. Our results are exact within the distant-observer approximation and completely independent of the nature of the tracers for which the correlations are evaluated. Focusing on 3-point statistics, we use an $N$-body simulation to study the joint probability density function of the relative line-of-sight velocities of pairs of particles in a triplet. On large scales, we find that this distribution is approximately Gaussian and that its moments can be accurately computed with standard perturbation theory. We use this information to formulate a phenomenological 3-point Gaussian streaming model. A practical implementation is obtained by using perturbation theory at leading order to approximate several statistics in real space. In spite of this simplification, the resulting predictions for the matter 3-point correlation function in redshift space are in rather good agreement with measurements performed in the simulation. We discuss the limitations of the simplified model and suggest a number of possible improvements. Our results find direct applications in the analysis of galaxy clustering but also set the basis for studying 3-point statistics with future peculiar-velocity surveys and experiments based on the kinetic Sunyaev-Zel’dovich effect.