  Let $\eta_t$ be a Poisson point process with intensity measure $t\mu$, $t>0$,
over a Borel space $\mathbb{X}$, where $\mu$ is a fixed measure. Another point
process $\xi_t$ on the real line is constructed by applying a symmetric
function $f$ to every $k$-tuple of distinct points of $\eta_t$. It is shown
that $\xi_t$ behaves after appropriate rescaling like a Poisson point process,
as $t\to\infty$, under suitable conditions on $\eta_t$ and $f$. This also
implies Weibull limit theorems for related extreme values. The result is then
applied to investigate problems arising in stochastic geometry, including small
cells in Voronoi tessellations, random simplices generated by non-stationary
hyperplane processes, triangular counts with angular constraints and
non-intersecting $k$-flats. Similar results are derived if the underlying
Poisson point process is replaced by a binomial point process.
