NFHS Softball has a specific ruling for a ball that has been caught and is then fumbled or deflected into dead-ball territory (Other adult softball codes probably have similar rules)...
NFHS 8-4-3g
A runner is entitled to advance without liability to be put out when:
g. the ball is in play and is overthrown (beyond the boundary lines) or is blocked...
EXCEPTIONS
1) When a fielder loses possession of the ball, such as on an attempted tag, and the ball enters the dead-ball area or becomes blocked, each runner is awarded one base from the last base touched at the time the ball entered the dead-ball area or became blocked.
Neither NFHS baseball or MLB has such a specific ruling that I can find for a caught ball that is fumbled into dead-ball territory.
The closest I can find for this scenario, is in Rule 7.
7.05 g and h
(g) Two bases when, with no spectators on the playing field, a thrown ball goes into the stands, or into a bench (whether or not the ball rebounds into the field), or over or under or through a field fence, or on a slanting part of the screen above the backstop, or remains in the meshes of a wire screen protecting spectators. The ball is dead.
(h) One base, if a ball, pitched to the batter, or thrown by the pitcher from his position on the pitcher's plate to a base to catch a runner, goes into a stand or a bench, or over or through a field fence or backstop. The ball is dead; APPROVED RULING: When a wild pitch or passed ball goes through or by the catcher, or deflects off the catcher, and goes directly into the dugout, stands, above the break, or any area where the ball is dead, the awarding of bases shall be one base. One base shall also be awarded if the pitcher while in contact with the rubber, throws to a base, and the throw goes directly into the stands or into any area where the ball is dead.
If, however, the pitched or thrown ball goes through or by the catcher or through the fielder, and remains on the playing field, and is subsequently kicked or deflected into the dugout, stands or other area where the ball is dead, the awarding of bases shall be two bases from position of runners at the time of the pitch or throw.
So the proper answer is one base in softball and two in baseball. (We expect more of baseball players
)
In your discussion you may want to refer to the deadball tables (Rule 5) in the NFHS rulebooks. Softball #15 (8-4-3g) and Baseball #15 (8-3-3c and 8-3-4). Also notice that there are 37 dead-ball situations in NFHS softball and only 26 in NFHS baseball. There are obviously some rules differences between the games... As noted by others here, THEY ARE NOT THE SAME GAMES.
[Edited by DownTownTonyBrown on Aug 11th, 2005 at 12:45 PM]