Surely that’d depend on the circumstances under which the flag was being transferred? If the original flagship is destroyed completely, which is a thing that can happen in war, then the new ship might be fishing the admiral and his staff out of the brine.
For sure. But what if it was damaged but not sunk?
Like in the song.
It happened a number of times in the Pacific War and generally what happened was that when the admiral needed to leave the ship, it was still in the chaos of the battle. They would have to leave quickly. It wasn’t practical to pull another capital ship in to transport the admiral so it would be done with small boats or maybe a destroyer.
Or by breeches buoy.
That happened in the Battle of Leyte Gulf. The Center Force command officer Vice Adm. Kurita had made his flag the heavy cruiser Atago and a destoyer had to come to the rescue. An American sub torpedoed it out from under him.
Dace made four torpedo hits on the heavy cruiser Maya (sister to Atago and Takao).[32] Atago and Maya both went down quickly with significant loss of life.[33] Atago sank so rapidly that Kurita was forced to swim to survive, though he was rescued by the Japanese destroyer Kishinami and then later transferred to the battleship Yamato.[34
Some of the staff were lost and it’s said that Kurita was tired by this experience, both of which may have contributed to his poor performance in the upcoming battle.
When they have time, they don’t have to swim.
In the Battle of Midway, the commanding officer Nagumo had the carrier Akagi as his flagship, and was forced to transfer his flag:
Realizing that his flagship was crippled, if not doomed, Admiral Nagumo decided to transfer his flag to the light cruiser Nagara, flagship of DesRon 10. The destroyer Nowake, therefore, was ordered to come alongside, and at 0746 Admiral Nagumo and his staff abandoned the Akagi’s bridge and prepared to board the destroyer. While this transfer was in progress the undamaged Hiryu, flying the flag of Rear Admiral Tamon Yamaguchi, ComCarDiv 2, launched her first attack against the Yorktown: eighteen dive bombers escorted by five or six fighters.
At 0830 Admiral Nagumo boarded the Nagara from the Nowake and promptly sent two radio dispatches, one to Admiral Yamamoto and one to the Mobile Force.
Note that there are three flagships here: Akagi, the initial flagship for the 1st Air Fleet, the force attacking Midway; Nagara, which was the flagship for DesRon 10 (10th Destroyer Squadron) then transferred his flag to it. Note that they still have one carrier left, but Nagumo, a black shoe admiral, decided he wanted a surface encounter and elected to transfer to Nagara. Hiryu was the flagship of the Second Carrier Division (with two of the four carriers).
I should expect so!