The purpose of a bankroll is to protect a long term winning player from going broke due to a run of bad luck. And for that reason I dont think, it makes any sense to have different bankrolls for different games or different
casinos. The more important consideration is to not automatically assume, you are a winning player, if you move to a new game, and play more conservative in the beginning, than what your bankroll allow.
The most classic mistake is people with experience from live poker moving to online poker and vastly underestimating, how much tougher online games are at a similar buyin level. Maybe they have been playing $1/$2 cash games in their local casino and build up a nice $10.000 bankroll. Then they take this online and get crushed in similar $1/$2 (200NL) cash games there. Or even $0,5/$1 (100NL) cash games.
Instead such a player should start at 5c/10c (10NL) games and play at least the first 50.000
hands there. Not for bankroll reasons but to build a track record proving, they are actually winning online. And also get used to the online environment like multitabling and table selection. So maybe such a player should set a goal of winning at least 30 BIs at each level before moving up. Meaning $300 at 10NL, $750 at 25NL, $1.500 at 50NL etc.
And if someone build their $10.000 bankroll online, it also makes sense to stick to the cheapest possible live games, until they have learned things like handling their chips, not giving away live
tells, and how live and online tendencies are different. So even though some would argue, that $10.000 is enough to take shots at $2/$5 live cash games, people should not be doing this, before they feel completely comfortable with the live environment.