Sometimes a twice-told-tale, or rather, a similar event occuring on parallel worlds of fiction, isn't an exact copy of one another. However, elements of each mirror the other in key areas, and such is evident in this team-up (or sorts) between Green Lantern and Batman.
On Earth-One, Bruce Wayne's manor is invaded by a criminal claiming he was framed yet proceeds to use science to achieve his amoral ends. At first duped into trusting the Time Commander, the capeless crusader is rendered immobile and seeking the help of ally Hal Jordan (instead of frequent chum Superman, which tips one off that something odd is going on in this case).
On Earth-Two, Alan Scott operates as his world's Green Lantern and lives in Gotham City, rather than Coast City where is counterpart frequents. Hence, his relationship with Bruce Wayne is much closer, and they've known each other for decades when this adventure begins.
Like his cosmic twin, Green Lantern is also drawn to Wayne Manor by a call for help. However, instead of a telegram (!?!) from the socialite as happened on Earth-One, an alarm is sounded. In this instance, Wayne is out of town to accept an honor as Batman (since most of his activities were in Gotham, could this mean he was in Metropolis where he often assisted his World's Finest buddy Superman?).
Yet again, a criminal has broken into the Manor on each world, only its GL who meets him rather than Wayne.
After revitalizing a seemingly comatose Batman and getting himself up-to-date on the home invasion, Lantern leaves for home only to be drawn back to Gotham after there is word that the dark knight has turned to crime for no good reason (he is rich, after all).
Now Hal suspects his new but "good" friend Bruce has become a bad seed, despite Batman's long and prestigous career that stated the contrary. Herein a hidden flaw is highlighted that GL shares with his counterpart, a paranoia bordering on the absurd towards his fellow man. Suddenly, Lantern is the all-knowing moral authority.
Such is the case with Alan Scott, who after cleaning up the mess at the Manor and in his own radio station's studio with some on air personalities, has to face a crime wave unprecedented which even leads to his own home being broken into!
Now frustrated with mankind in general, Green Lantern uses the mighty energies of his power ring in an unprecedented way, wishing to exile all evil from his world. The ring obliges, and all humans including GL himself are wisked away from Earth-Two leaving a barren globe also left without justice and the good in man!
Hal and Bruce finally get to the bottom of Batman's turn to the dark side. In actuality, it is a disguised Time Commander, empowered by Lantern's own energy and causing temporal chaos to ensue. Threatening Gotham City's survival unless Commisson Gordon clears the villains civilian identity (?!?), it's up to our pair of heroes to face off against this foe.
With one sent a day in the future and the other a day in the past, only thanks to the "timely" aid of GL's power ring are the duo able to coordinate their efforts and ultimately defeat the Commander. At least until later that year when he returns one more time.__________
And in our second tale, Hal is able to ascertain that the mass exodus of people from Earth-Two, now residing immobile in the salt plains of Nevada on his Earth-One, was due to Alan Scott's well-meaning but ill-advised intervention on community ills.
Thanks to the will power and rings of the Lanterns Green, all of Alan's civilization is transplanted back to Earth-Two, wherein Hal and Alan are able to track the source of the crimewave to a radio talk personality under Alan's employee that doubles as a gang leader. And all learn a valuable lesson, humans have to wrestle with what is good and what is evil continually.