  We study $S_N$-invariant four-point functions with two generic multi-cycle
fields and two twist-2 fields, at the free orbifold point of the D1-D5 CFT. We
derive the explicit factorization of these functions following from the action
of the symmetric group on the composite multi-cycle fields. Apart from
non-trivial symmetry factors that we compute, the function with multi-cycle
operators is reduced to a sum of connected correlators in which the composite
fields have, at most, two cycles. The correlators with two double-cycle and two
single-cycle fields give the leading order contribution in the large-$N$ limit.
We derive explicit formulas for these functions, encompassing a large class of
choices for the single- and the double-cycle fields, including generic Ramond
ground states, NS chiral fields and the marginal deformation operator. We are
thus able to extract important dynamical information from the short-distance
OPEs: conformal dimensions, R-charges and structure constants of families of
BPS and non-BPS fields present in the corresponding light-light and heavy-light
channels. We also discuss properties of generic multi-cycle $Q$-point functions
in $M^N/S_N$ orbifolds, using a technology due to Pakman, Rastelli and Razamat.
