  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 $\epsilon x$ at time
$(\epsilon\nu([\epsilon,1]))^{-1}$ behaves like
$\epsilon^{-1}\lambda\_1(]0,x[)$ as $\epsilon\to 0$, 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.
