Except in very serious cases, "the crime dies with the clone".
(I'll be working from the Paranoia XP edition here, but most of this advice should hold for all Paranoias.)
Most of those very serious cases are having a mutation dangerous enough that your cell line can't be allowed to continue, like Machine Empathy. For everything else? Well, suppose Bob-R-GON-1 backs Alice-R-DED-1 into a rhetorical corner, after which Alice decides she might as well go loud and breaks out her Electroshock mutant power, shouting "PURGE DESTROYS ALL!" After which her fellow Troubleshooters execute her for treasonous acts, as is right and proper.
Does everyone now know that Alice-R-DED-2 is a PURGE sympathizer with Electroshock? Well, the players probably do, because all Alice did was tick up a number on her character sheet and wait, but:
Likewise, if you terminate Martin-R-TSD-1 for having the mutant power of Slippery Skin, don't start firing immediately when Martin-R-TSD-2 arrives. You, the player, may know his skin is still slippery, but your character doesn't. Making every clone identical is a game convenience; it saves a lot of time creating new characters, and is the basis for some cheap jokes. However, the idea that all backup clones have the Prime's mutant power is unknown in Alpha Complex or to The Computer.
-- Paranoia XP, p.39, "Etiquette"
The "likewise" refers to the memories Alice-R-DED-2 has of her previous clone's final moments and the grudge she bears against Bob-R-GON-1 for putting her in that no-way-out position. Just jumping out of the clone pod guns blazing might be fun for a few seconds, but a slower-burn I-know-that-you-know-that-I-know bit of jockeying can last several satisfying sessions before Bob-R-GON-1 yelps from a sudden electric shock and then screams as he falls into an active food vat. And then Alice has no grudge against Bob-R-GON-2, but there's probably one coming the other direction. And so it goes.
Assassination directives work in the same way. If someone else in Alpha Complex thinks Bob-R-GON-1 deserves to die, they will have reasons why that relate to specific things that Bob-R-GON-1 has done, not anything the -2 through -6 are doing. Depending on your version of Paranoia, 2 through 6 may just be motionless clone blanks waiting for 1 to bite it.
...unless it would be ironic, or funny, or ironically funny.
Of course, there are all the non-murder reasons you might want to accost someone, and for a lot of those their clone number doesn't particularly matter. Tongueprints and credit accounts will usually just drift from clone to clone. But maybe someone's done something so heinous that, in a secret society's eyes, it's the equivalent of getting Machine Empathy, and they want a scalp and don't care about the number.
Or maybe an innocent clone blank just isn't working out at the table. In which case, well:
A mutant just got bumped off and poured into his waiting clone backup. Does he have the same power? If he previously registered his mutation, is he still registered? If not, can he, or does he have to, register again?
Groan. These annoying questions arise from thinking too hard about PARANOIA's clone backup system -- a system, we repeat, that exists solely as a story device so players can keep playing without having to generate a new character every two minutes. But your players will ask these questions, so it falls to us as Famous Game Designers to answer them.
The answer is, 'We don’t know.'
-- Paranoia XP, p.151, "Mutant Powers"
This isn't to completely destroy the idea of the blank-slate clone. That's still the assumption players are encouraged to make. The GM explicitly gets final authority on the matter, but that's a call that varies from table to table with the GM's sense of what will work for their playgroup, and not really something I can answer to a blank Infrared room on the Internet.
