  Birkner et al. obtained necessary and sufficient conditions for the frequency
between two independent and identically distributed continuous-state branching
processes time-changed by a functional of the total mass process to be a Markov
process. Foucart et al. extended this result to continuous-state branching
processes with immigration. We generalize these results by dropping the
independent and identically distributed assumption. Our result clarifies under
which conditions a multi-type $\Lambda$-coalescent can be constructed from a
multi-type branching process by a time change using the total mass. Finally, we
address a problem formulated by Griffiths, by clarifying the relation between
2-type $\alpha$-stable continuous-state branching processes and 2-type
$\beta$-Fleming--Viot processes with mutation and selection.
