I would like some advice from someone who has done this (if it will work) and I am particularly interested in how Chris Kern runs his as his April post was the one that got me pointed in the right direction.
I do pretty much as John Beardsworth suggests, with one minor tweak. I have only one permanent catalog[ue]; I've never found a reason to maintain more than one. When I want to process files on a machine that does not have access to it, I export the files I plan to work on
as a catalog, copy the exported catalog to the target machine, and import it into the instance of Lightroom on the target. I subsequently update the source machine with any changes by reversing the process. In other words, I treat the catalogs on all but one machine as temporary repositories, and delete the transient contents after I have migrated any changes to the permanent catalog.
When traveling — i.e., when I don't have access to the permanent catalog — I create a temporary catalog on a laptop for new captures. When I get home, I export the image files and edits, again as a catalog, and import them into the permanent catalog.
This two-way export-import process is a bit clunky, to be sure, but it's not unduly burdensome once you are accustomed to it. And Lightroom isn't designed with the mechanisms that would allow you to run multiple instances on different computers and update a single catalog asynchronously.
I haven't encountered any issues moving exported catalogs between the Lightroom subscription (CC) and static (6.X) products.