We relate finite generation of cones, monoids, and ideals in increasing chains (the local situation) to equivariant finite generation of the corresponding limit objects (the global situation). For cones and monoids there is no analogue of Noetherianity as in the case of ideals and we demonstrate this in examples. As a remedy we find local-global correspondences for finite generation. These results are derived from a more general framework that relates finite generation under closure operations to equivariant finite generation under general families of maps. We also give a new proof that non-saturated Inc-invariant chains stabilize, closing a gap in the literature.
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