I think that one character at a time would be sufficient, but if we are worried about deaths then we could pre-plan to have three characters in the wings to take over. Just like a DnD campaign, your character dies? Oops, oh look here is your cousin Bob here to take over. But also, this leads to another meta-gaming situation. If someone's character is killed, it would presumably be done by the culprit or associated with them. We could all read said murder, and we could then act off of that, whether our characters saw it or not.
I personally would stick with one character and one character only, makes it a much more risky situation. You have to survive and detect smartly, you know you are on the trail of a murderer you can't make any stupid mistakes and jump to any wild conclusions without proof. If they all die before figuring out who the killer is, well then, the murderer gets away with it. A realistic scenario, makes a way for the murderer to get away with it and 'win' too.
Beyond this, are all the players private detectives that have been brought in by the local governance to help out the constabulary, sleuthing for our own personal vendettas, or are we all working for the constabulary? Or shall we make this just a part of our individual characters backgrounds?