Who is allowed to kneel influence to move it to another character? The player whose character it is attached to? The player who originally played it? Anyone at the table? And do you have to attach it to another character you control, or can you attach it to any character on the table?

The attachment is controlled by the person who originally played the attachment, it's cost and activation can only be done by it's controller. Also, You can attach it to any character as long as that character does not have the "No attachments" ability.

Thanks.