If you pick up the parameters of a label, it should store the text settings exactly. This includes the Text Block Frame and the text within it. Try Alt/Opt clicking on a multiline label, then going to the Default Label Settings dialog. Under the Text panel, you should see the lines of text exactly as they were picked up, although you are able to change it here if you want.
It is a 'feature' that injecting from one label into another, won't transfer the actual text. Note also that the geometry is never transferred, not for any element.
Cheers,
Link.