Yes the modification is not carried over and the cross-post will often have a different author from the original too. While creating the cross-post the cross-poster could have edited the title and description, too. The only connection is that they have the same url (in the attachment). Lemmy adds the text "cross-posted from link-to-original" to the beginning of the body content but it's just text, not an attribute on the Activity.

Room for improvement!

Also moving what NodeBB calls a topic (a post in Lemmy/PieFed) from what NodeBB calls a category (community in Lemmy/PieFed) into a different category (without spawning a new topic or leaving the old copy behind) is much needed but not implemented in Lemmy/PieFed/Mbin.

Cross-posts are entirely new and separate posts from their originals.

You could save yourself some hassle and make urls and IDs the same value. But you do you

Object IDs should really be retrievable by http GET. See section 3.2 of https://www.w3.org/TR/activitypub