  Compartmentalised biochemical reactions are a ubiquitous building block of
biological systems. The interplay between chemical and compartmental dynamics
can drive rich and complex dynamical behaviors that are difficult to analyse
mathematically -- especially in the presence of stochasticity. We have recently
proposed an effective moment equation approach to study the statistical
properties of compartmentalised biochemical systems. So far, however, this
approach is limited to polynomial rate laws and moreover, it relies on suitable
moment closure approximations, which can be difficult to find in practice. In
this work we propose a systematic method to derive closed moment dynamics for
compartmentalised biochemical systems. We show that for the considered class of
systems, the moment equations involve expectations over functions that
factorize into two parts, one depending on the molecular content of the
compartments and one depending on the compartment number distribution. Our
method exploits this structure and approximates each function with suitable
polynomial expansions, leading to a closed system of moment equations. We
demonstrate the method using three systems inspired by cell populations and
organelle networks and study its accuracy across different dynamical regimes.
