  $\Lambda$-Wright--Fisher processes provide a robust framework to describe the
type-frequency evolution of an infinite neutral population. We add a polynomial
drift to the corresponding stochastic differential equation to incorporate
frequency-dependent selection. A decomposition of the drift allows us to
approximate the solution of the stochastic differential equation by a sequence
of Moran models. The genealogical structure underlying the Moran model leads in
the large population limit to a generalisation of the ancestral selection graph
of Krone and Neuhauser. Building on this object, we construct a continuous-time
Markov chain and relate it to the forward process via a new form of duality,
which we call Bernstein duality. We adapt classical methods based on the moment
duality to determine the time to absorption and criteria for the accessibility
of the boundaries; this extends a recent result by Gonz\'alez Casanova and
Span\`o. An intriguing feature of the construction is that the same forward
process is compatible with multiple backward models. In this context we
introduce suitable notions for minimality among the ancestral processes and
characterise the corresponding parameter sets. In this way we recover classic
ancestral structures as minimal ones.
