This article discusses a combinatorial extension of tropical intersection theory to spaces given by glueing quotients of partially open convex polyhedral cones by finitely many automorphisms. This extension is done in terms of linear poic-complexes and poic-fibrations, mainly motivated by the case of the moduli spaces of tropical curves of arbitrary genus and marking. We define tropical cycles of a linear poic-complex and of a poic-fibration, and discuss the pushforward maps in these situations. In the context of moduli spaces of tropical curves, we also discuss "clutching morphisms" and "forgetting the marking" morphisms. In a subsequent article we apply this framework to moduli spaces of discrete admissible covers and study the loci of tropical curves that appear as the source of a degree-$d$ discrete admissible cover of a genus-$h$ $m$-marked tropical curve, for fixed $d$, $h$ and $m$.