It sounds like you have multiple linked record fields joining two tables. Yes, it is possible to have automations move a linked record from one linked field to another, as long as both fields point to the same table.
You will need to have formula fields and rollup to identify which linked records need to be moved. Then you need the actions to make the changes.
If each appointment links to only one client, you could use the update action to remove the link from one linked record field and add the links no to another field.
If you have a more complex situation, you might be better off using a script to handle the logic regarding which linked record fields to use.
I figure it out:
I’ve got the A client record links to several appointment records. Then I have a lookup field that looks up the linked appointments and filters them “where Appointment is before Today”. Then an automation which triggers when the lookup field is updated, and copies the lookup field contents into a linked record field called “History” which turns all the lookup info into linked records.
Not sure if I explained that clearly, but it works.
[I found some advice from ScottWorld on another post that clued me in to how to automatically link records by pasting the primary field of a record into the linked record field. That was the missing link]
Thanks for the help.