  We propose a general framework for the study of the genealogy of neutral
discrete-time populations. We remove the standard assumption of exchangeability
of offspring distributions appearing in Cannings' models, and replace it by a
less restrictive condition of non-heritability of reproductive success. We
provide a general criterion for the weak convergence of their genealogies to
$\Xi$-coalescents, and apply it to a simple parametrization of our scenario
(which, under mild conditions, we also prove to essentially include the general
case). We provide examples for such populations, including models with
highly-asymmetric offspring distributions and populations undergoing random but
recurrent bottlenecks. Finally we study the limit genealogy of a new
exponential model which, as previously shown for related models and in spite of
its built in (fitness) inheritance mechanism, can be brought into our setting.
