I am interested in learning about limits with respect to ultrafilters on a poset (partially ordered set) and ultraproducts with respect to a poset. However, all I can find about this topic only consider ultrafilters on the natural numbers (more precisely, ultrafilters on the power set $\mathcal{P}(\mathbb{N})$ with partial order given by set inclusion). For example, see these wikipedia articles and this blog by Terrance Tao.
https://en.wikipedia.org/wiki/Ultralimit
https://en.wikipedia.org/wiki/Ultraproduct
Does there exist a theory for general posets? If so, it would be much appreciated if a brief description could be provided/point to any relevant references.
Wikipedia does treat filters on general posets. But more interestingly: prove/check the Cayley-type result that every poset is a sub-poset of some powerset ordered by inclusion