In the Las Vegas casino's (years ago), each cardroom that I worked in was a 24/7 operation. Security would take the drop box and it would go to the casino count room. Each box was marked; poker, table # and shift:
POKER GRAVE
TABLE # 3
Security would start at about 15 minutes before the next shift began, There were two security personel for each procedure, one table at a time. The removed box would be replaced by the empty box for the next shift. I imagine, with the larger cardrooms of today, more than one team of security guards could be used.
Other rooms have their own count room and dealers stay after their shift and count the drop. As far as I know, the bad beat jackpot could be handled in the same manner as the drop. The regulations governing these procedures may vary from casino to casino. There will always be multiple employees present, from different departments for confirmation.