We prove several limit theorems that relate coalescent processes to
continuous-state branching processes. Some of these theorems are
stated in terms of the so-called generalized Fleming-Viot processes,
which describe the evolution of a population with fixed size, and are
duals to the coalescents with multiple collisions studied by Pitman
and others. We first discuss asymptotics when the initial size of the
population tends to infinity. In that setting, under appropriate
hypotheses, we show that a rescaled version of the generalized
Fleming-Viot process converges weakly to a continuous-state branching
process. As a corollary, we get a hydrodynamic limit for certain
sequences of coalescents with multiple collisions: Under an
appropriate scaling, the empirical measure associated with sizes of
the blocks converges to a (deterministic) limit which solves a
generalized form of Smoluchowski's coagulation equation. We also
study the behavior in small time of a fixed coalescent with multiple
collisions, under a regular variation assumption on the tail of the
measure $\nu$ governing the coalescence events. Precisely, we prove
that the number of blocks with size less than $\varepsilon x$ at time
$(\varepsilon\nu([\varepsilon,1]))^{-1}$ behaves like
$\varepsilon^{-1}\lambda_1(]0,x[)$ as $\varepsilon\to 0$, where
$\lambda_1$ is the distribution of the size of one cluster at time $1$
in a continuous-state branching process with stable branching
mechanism. This generalizes a classical result for the Kingman
coalescent.