Multi-particle cumulants are a widely used experimental method to disentangle few-body short-range azimuthal correlations, usually called non-flow, from many-body long-range ones, which carry information on collective dynamics. Cumulants can be defined in terms of correlators, which can be efficiently computed from $Q$-vectors via the Generic Framework (GF) algorithms, especially when direct implementation of analytic formulae is unpractical. One limitation of the GF algorithms is that correlators must be computed from one common set of particles, whereas in some cases one would wish to correlate particles from disjoint or intersecting subsets. For some of those cases, analytic solutions have been previously derived, e.g. the so-called differential cumulants and subevent cumulants. Here, we present a generalization of the GF algorithms which enables to automatically compute $n$-particle correlators between $n$ or less arbitrary subsets, with $n=2, \ldots\infty$, without the need of lengthy analytic formulae. We discuss possible applications and limitations of cumulants built from such correlators, taking as a case study the Chiral Magnetic Effect.