Mysteries, by the nature of their form, must contain a great deal of exposition, that is, information which must be passed on to the audience through speech. Often this information transfer is done directly, when one suspect comes right out and tells another suspect or a detective something, purely for the information value: "But you realize, Inspector, that Mister Jones was left handed, and would not have stabbed Mister Smith in the back at that angle."
Sometimes, this can be the only way to get a fact across, but if information transfer is the only thing going on in a scene it can get dull rather quickly. And we've all seen scenes where, in a rush to get information to the audience, the characters are made to tell each other what they already know.
"Well, Jonathan B. Metzenhimer! Billy's son! I haven't seen you since you ran
off to Africa with the Teetsee Fly Expedition of 1925 with my estranged sister Mabel, the one with the glass eye and the wooden leg with a termite problem, where you were attacked by savages and forced to live as the world's tallest pygmy goatherd until your amnesia cleared up. How's that malaria problem?"
Better Jonathan should have a reason to tell the story himself to someone who didn't know it, thus gaining the full emotional impact of the story, or at least a chance to milk its comedic potential, without making the teller sound like a babbling idiot.
There should be some sort of conflict at the spine of every scene, generated in the clash of one character who wants something, and another who wants to prevent that character from reaching that goal. We can still be fed the information in the scene, but it is our interest in the resolution of the conflict that will hold our attention.
For example, in the scene between Superthud and Sarasota in The Dying Old Man On the Flying Trapeze, Superthud wants to marry her, but she objects. In the course of the scene we learn Superthud's motive for killing her father and his knowledge of the lion's violent reaction to Chanel #2 perfume, yet the dramatic thrust of the scene is will he or won't he convince her to marry him? The facts come out as just another part of the conversation.
In the heat of emotion a killer might blurt out their motive when, had they their wits about them, they would have kept their mouths shut. They might brag about some point of the killing to someone they trust, or inadvertently let slip some knowledge vital to the crime in a context that has nothing to do with the murder. What they should not do is lecture.
You may also allow the audience to question the suspects directly, after enough preliminary information has been presented to give them something to ask about. If questions are allowed the entire cast must be well versed in the complete mystery story, so as to eliminate mistakes on their parts which may be interpreted as clues.