Lightroom will store all the edits in the Lightroom catalog itself, so just copying the sidecar won't work. There is a method to tell Lightroom to write metadata to each file, but I am not 100% sure how much of that will get put into the sidecar and if that would work. I would suggest experimenting on a few copied files and seeing for yourself if it copies over what you need, or asking in the Adobe forums
In Photo Mechanic 6 you can find all images in a Contact Sheet that have empty caption fields.
Bring up the Find window (cmd-f on macOS)
Select "All Items" from the dropdown
Select the checkbox to search Metadata
Select only the checkbox to search Caption
Leave the Search field blank.
This will Select the items in that Contact Sheet with empty caption fields. Press F2 to filter the Contact Sheet to only show Selected items if needed.
If you are using Photo Mechanic Plus, you can search your entire catalog(s) at once with the search string "caption is empty"
I hope this helps,
Mick
