Just to be clear, I want to know if there's any chance I could let users move threads of specific forums but nothing else. Not the whole moderation panel, because I don't want them to edit other people's post/threads, only move them. Is this possible? Thanks in advance.
This would require a plugin, the ModCP has some features that can be disabled, but many of them cannot, as such you wouldn't be able to allow them just to move posts, they would also have access to the ModCP. Inline thread moderation is only available to those with Moderator and Administrator privileges, and this would be the only way for them to move posts without ModCP access.
I hope we can do this by allowing mod permission ( only move ) to desired groups. I haven't tried that, certainly worth checking.