Backport various fixes from !3479 (fix linking with measure LPE, fix multiple copy-pastes, fix copy-paste clones to new document)

Backports part of Jabier Arraiza's fix from a5f8da8b

Fix #387 (closed), #1743 (closed), #3035 (closed)

/cc @jabiertxof could you review?

Update: Jabier's taken a quick look at it, but he's pretty busy too. From DMs:

I see your code seems tiny and correct,