For the swag to be important enough to potentially spark a mob war, the corp has to be somewhat in bed with one of the mobs. But, before we get into that, we have to clarify some things.
Organized crime syndicates aren't necessarily monolithic. Each faction has many smaller factions within them, and even they are not fully unified, and there's plenty of infighting all around. So, to have the Yakuza blamed for, say, a run made on a corp facility protected by the Mafia, you have to be a bit specific on who's family is being hit and by which faction of the Yakuza. But, this in and of itself may not really spark a mob war.
Because what really sparks a mob war is when it gets personal. Sure, having one mob hit another mob's racket that's set up might do it, but chances are all that's going to result in is a minor retaliation. However, kill a Made Man, or even a Mafia Boss, and have the Yakuza blamed for that, and you have the potential for a war. What's then essential is to make sure that both sides don't get together and determine the truth, which could be part of the adventure itself.
Immediately following a hit on a Made Man or a Mafia Boss is that the Mafia will talk to the other smaller Mafia families in the territory and deals may be made in order to coordinate a response. What's important here is how powerful each faction is. It may be that the Yakuza may be more powerful than the Mafia, and therefore the big boss won't want an all-out war, in spite of the loss of one of theirs and the indignity that comes from that. Besides, it'll be bad for business, and even if the Mafia should win, they're going to be hard-pressed to recoup their losses.
So you can have your players do a run on a corp facility protected by the Mafia, or Triads, or whatever, and in the process of stealing something, they kill someone important, and maybe that was intended all along by Mr. Johnson. If the PCs don't do it on their own, have Mr. Johnson shadow them with another professional runner, a hitman who will use the run to create just enough of a distraction in order to kill the real target.