  For a class of $\Lambda$-Fleming-Viot processes with Brownian spatial motion
in $\mathbb{R}^d$ whose associated $\Lambda$-coalescents come down from
infinity, we obtain sharp global and local modulus of continuities for the
ancestry processes recovered from the lookdown representations. As
applications, we prove both global and local modulus of continuities for the
$\Lambda$-Fleming-Viot support processes. In particular, if the
$\Lambda$-coalescent is the Beta$(2-\beta,\beta)$ coalescent for
$\beta\in(1,2]$ with $\beta=2$ corresponding to Kingman's coalescent, then for
$h(t)=\sqrt{t\log (1/t)}$, the global modulus of continuity holds for the
support process with modulus function $\sqrt{2\beta/(\beta-1)}h(t)$, and both
the left and right local modulus of continuities hold for the support process
with modulus function $\sqrt{2/(\beta-1)}h(t)$.
