Disagree,

I flew FCO->MXP on LH and continued on MXP->EWR->GRR.

When I left FCO my bag was tagged with a LH bag tag that indicated VIP, but no priority tag other than that. When I arrived at GRR, someone along the way had opened up the LH tag and added a CO Priority tag in the LH tag.

Needless to say I was pretty shocked, and my bag was the third bag off the plane at GRR.

Your milage might vary, but they can re-tag it mid-trip!

-Chris