Not to be confused by the game of the same name, a dying earth campaign should be set in a world on its last legs. The characters are the last generation of heroes, giving the world either a bleak [Sin City: Last Goodbye (Marv's story)/Wild Bunch/Last Samurai] tone to it OR a very postmodern [Black Company, Cowboy Beebop, Joss-anything] tone.
Setting: The game world of your choosing undergoes a radical, but not apocalyptic change. Magic fades; technology advances at an exponential rate; orcs are wiped out; dungeons are completely looted; the world is mapped; a civil war devastates both nations in war; the gods leave; monotheism sweeps the land; the king outlaws gold-chasing; etc.
The event triggers a new era and adventurers are a dying breed. There's no need for people who go into the frontier and clear out all the goblins, because... well... the frontier is gone.
Rules: Humans and Elves are likely to be the only playable races and things likes Barbarians and Druids are near extinct. Playing one makes little sense. If magic is dying, wizards are done too. Depending on what is bringing about the "death of this world," though... the rules will change. Bear in mind that Rangers cannot exist in a world that must "industrialize" the process of logging, hunting, farming, and herding. Here's a matrix Kevin Wilson came up with once.
I think it illustrates pretty well some of the anachronistic thinking that goes into D&D. Paladins are a luxury in a primitive society, but rote in a modern (Renaissance-y) era game. But to have Paladins and Barbarians juxtaposed together suggests that whatever game world they are devising is either... backwards or poor-planned... can you imagine a world where England doesn't exploit India?
That's the Forgotten Realms.
World Logic: Corruption comes shortly after a society evolves into an "industrialized state." People eager to make more coin will join one of two sides in the war of greed... get into the pocket of politicians and become Lawful Evil... or bribe cops so your Guild can continue operating "legally" and become Lawful/Neutral Evil.
Missions: Adventures in this world are not railroaded, nor are they planned very well. The PCs are scrounging for work, looking to eek out a life in whatever corner of the world they can. Maybe there's a lawless "Mexico" that's less than 5 years from being modernized or perhaps there's a boat headed for somewhere... but the typical kill the orc, take his pie adventure has been done by 51 generations before you.
There are no pies left.
DM Advice: Keep the campaign focused on the PCs. Everything is about their personal struggles. In fact, it can make the adventure more interesting if the PCs don't even like or respect each other... but are forced together because they have to. If this is a single adventure or short-term campaign, the PCs should be put into a no-win situation... ala Wild Bunch. If the Dying Earth Campaign is really a campaign, then the PCs should each be thrown into a situation where they are desperately trying to hold onto old ideals (ala Star Wars: A Hew Hope) and refuse to go down swinging.
In this latter example, the PCs can actually win, and throw the world back into the dark ages if they want... which might be better than a world with Bards.