Hi @akshatas5875550,
Could you please let know your AEM version and how you performed "Move" operation?
Per AEM 6.5.0, it works fine.
"Move" action on specific content fragment will list the referencing page to adjust/republish the new path there accordingly. By this way, content fragment won't be removed from pages where it is referred (Usual OOB Move behavior like we have for site pages)