  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.
