Broad question, no formulas: are there reasons why Gaussian Processes should emerge from natural phenomena in the first place?
For individual Gaussian Random Variables there are multiple versions of the central limit theorem that provide reasons for their existence. Furthermore, I have seen some attempts at explaining why Gaussian distributed observations emerge from phenomena that do not involve an uncountably large number of parent events.
But is there such a thing for Gaussian Processes? I often see GPs used as an artificial construct used to model data, and way too often I see the sentence "GPs are a popular technique used in machine learning", usually followed by a list of virtues of this technique. But I never encountered "This process has this and that properties and as such it should be modelled as a GP".
So, back to the original question, any reason for a natural process exhibiting features that justify it being modelled as a GP?