I have never had it happen but I would think it would go as follows
Say you have a 34" perm registered AMHR horse, take it to a show, measures 34.5" (all three times) - (taking it this would be this horses first show as a 3yr past horse) it would then have that measurement sent to the office and receive a permanent card back in the mail (as any 3yr past horse would)
Then you could either send the papers back to have the height changed, or next time you sent the papers in, say for transfer ext that "permanent" card height that the office printed and sent to you, that height would be put on this horses papers.