Moving Data Products Between Governance Domains

I have a number of data products whose location in the governance domain hierarchy is no longer a good fit.

I'm struggling to find a route to move them, unpublishing and select edit gets me to the same options as during creation but the governance domain can't be selected and change.

https://learn.microsoft.com/en-us/purview/how-to-create-manage-data-products#edit-data-product is ambiguous on what can be edited but there's no exclusions given. Copilot makes a number of suggestions (dragging and dropping and using the "move" option from the list of data products), unfortunately none of of these appear to be based in reality.

Is there a route to make a move? Am I missing permissions? Or is the expectation that I delete and re-create data products any time they need to move.