  Finding the most powerful node in a dynamic random network, the largest set
in a partition-valued stochastic process, or the largest family in an evolving
population at a given time, can be a very difficult problem. This is
particularly the case when the underlying stochastic process has complex
dependencies and the individual strength of an object has an impact that only
plays out over time. We propose a novel technique to deal with such problems
and show how it can be applied to a broad range of examples where it produces
new insight and surprising results. The method relies on two steps: In the
first step, which is highly problem dependent, the problem is embedded into
continuous time so that the evolution of the sizes of objects after their
individual birth times become approximately independent while we only need
minimal control over the birth times themselves. Once such an embedding is
achieved, the second step is to apply a Poisson limit theorem that allows a
comparison of object sizes in a critical window and therefore allows a
description of features of extremal objects. In this paper we prove such a
versatile limit theorem, based on extreme value theory, and show how the
technique can be used to study extremal behaviour in different types of
preferential attachment networks with fitness, branching processes with
selection and mutation, and random permutations with random cycle weights.
