Answer

The fewest possible moves for getting the prisoners into their dungeons in the required numerical order are twenty-six. 

The men move in the following order:—1, 2, 3, 1, 2, 6, 5, 3, 1, 2, 6, 5, 3, 1, 2, 4, 8, 7, 1, 2, 4, 8, 7, 4, 5, 6. 

As there are never more than one vacant dungeon to be moved into, there can be no ambiguity in the notation.

The diagram may be simplified by my "buttons and string" method.
It then takes one of the simple forms of A or B, and the solution is much easier.

In A we use counters; in B we can employ rooks on a corner of a chessboard. 

In both cases we have to get the order

in the fewest possible moves.