  When two (possibly different in distribution) continuous-state branching
processes with immigration are present, we study the relative frequency of one
of them when the total mass is forced to be constant at a dense set of times.
This leads to a SDE whose unique strong solution will be the definition of a
$\Lambda$-asymmetric frequency process ($\Lambda$-AFP). We prove that it is a
Feller process and we calculate a large population limit when the total mass
tends to infinity. This allows us to study the fluctuations of the process
around its deterministic limit. Furthermore, we find conditions for the
$\Lambda$-AFP to have a moment dual. The dual can be interpreted in terms of
selection, (coordinated) mutation, pairwise branching (efficiency),
coalescence, and a novel component that comes from the asymmetry between the
reproduction mechanisms. In the particular case of a pair of equally
distributed continuous-state branching processes, the associated $\Lambda$-AFP
will be the dual of a $\Lambda$-coalescent. The map that sends each
continuous-state branching process to its associated $\Lambda$-coalescent
(according to the former procedure) is a homeomorphism between metric spaces.
