  A class of Fleming-Viot processes with decaying sampling rates and
$\alpha$-stable motions that correspond to distributions with growing
populations are introduced and analyzed. Almost sure long-time scaling limits
for these processes are developed, addressing the question of long-time
population distribution for growing populations. Asymptotics in higher orders
are investigated. Convergence of particle location occupation and inhabitation
time processes are also addressed and related by way of the historical process.
The basic results and techniques allow general Feller motion/mutation and may
apply to other measure-valued Markov processes.
