Your title question is very broad, but as-written it seems restricted to "how do I plumb between two places rotating with respect to one another?"
There's a rotating house in La Mesa, CA that solves this problem. Inventor/first owner Al Johnstone has at least a couple patents on it including US703235B2 "Swivel joint apparatus and method for utility supply to a rotatable building" and Tom Scott has done a video on it. The central shaft is immobile and connects the outside utilities to cutouts. Those cutouts are sealed against the moving side of the house into annular chambers, which the moving side also taps into.
The video has wonderful illustrative graphics and Al Johnstone does a fine job explaining the apparatus in the interview. I feel more secure only reproducing the patent images here.
The rotating outer jacket:
The inner, nonrotating service to the annular chambers:
There are other conceivable schemes to move heat across a rotating boundary; presumably you could use conduction between bits of metal with suitable grease. But there's more than one reason to want to plumb across a rotating boundary anyway, and I think some version of this will be what folks would have to use to do it.
Per Woody's answer the plumbing portion of this seems to be essentially just a larger, adapted version of the "hydraulic swivel manifold." I typically avoid Youtube comments but I see commenters on the Tom Scott video also mentioned that (but not by name) and also applications in automatic transmission gearboxes. Also seems to be known as a rotary union (that has a very good cutaway depiction). I'm curious how far back the invention goes but haven't found that yet.
It might be better to have this asked as a separate question so people could find this information more easily, but I've searched NTRS for the list of terms equivalent to rotary union from the relevant Wikipedia article and included whatever I've found that seems relevant. "Rotary coupling" and "rotary joint" are the only terms of all of them that seemed to generate relevant results.
"Rotary coupling" includes "NASA Tech Briefs" Rotary Coupling Extends Life Of Hose
Conceived for use on Space Station to transfer vapors across rotary joints to directional radiators for condensation or to transfer liquids to gimballed payloads for evaporation
Rotary Fluid Coupling. I stumbled across the actual tech brief (which is in a magazine?) somehow while looking for a different one.
Rotary coupling for heat-transfer fluid contains four lines for vapor and four corresponding liquid-return lines
and (I'm guessing presented during the Space Station Freedom design effort--ID is prefixed with 1986?) conference paper A steerable radiator for spacecraft application
Future large space structures such as the Space Station will have high dissipation and long life requirements which dictate the requirements for steerable radiators. Several rotary coupling concepts were considered to accomplish heat transport across the steerable radiator system's rotating interface. Rotary fluid couplings were chosen over rotary contact couplings or flexible lines because of low temperature gradients and operational flexibility.
"Rotary joint" has the Tech Brief Rotary Joint for Heat Transfer
Rotary joint exchanges heat between two heat pipes - one rotating and one stationary. ... wicks in central artery of heat pipe separate into multiple strands that lead to concentric channels on rotary interface side of stationary disk. Thin layer of liquid sodium/potassium alloy carries heat from one member of rotary joint to other. Liquid conducts heat efficiently while permitting relative motion between members. Polypropylene rings contain liquid without interfering with rotation.
So at least some actual engineering thought/work went into rotary unions for radiators on US space stations. I haven't yet tried to look into why the current ISS radiators aren't plumbed this way.