Let R be a standard graded polynomial ring and let I be a homogenous prime ideal of R. Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of R/I^t as t grows arbitrarily large. I will discuss their results and give an explicit description of the transition maps between these local cohomology modules in a particular example.
I will also consider asymptotic structure in a different direction: as the number of variables of R grows. This study of families of modules over compatible varying rings hints at the existence of OI structures, which I will discuss if time permits.

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