Constantine's "birth date" must just be a reference to his first actual appearance in Swamp Thing.
Last issue, John Constantine killed Earth-2 John Constantine because that's the closest he'll ever be able to come to killing himself. Except for that time he killed himself when he had lung cancer but that doesn't really count because he knew the bureaucracy and red tape to recover his soul would be so confusing that nobody would allow him to die. So that was less a suicide attempt and more just being a Smart Alec.
Constantine also opened a gate between Earth-2 and Earth-Main-Earth so he could finally return home after his relatively boring trip into the past. He was hoping to take along his lost family and friends but after murdering their John right in front of them, he's kind of lost them as well. But Doctor Cocoa Puffs is still trying to hitch his oxen to the wagon that he's hoping to hitch to Constantine's star because he wants to stop being Doctor Cocoa Puffs and once again become good old normal Doctor Fate. But I'll only ever call him that when he resides on Earth-Main-Earth! So he'd better come up with a spell that allows him to follow John! This story probably takes place after Doctor Cocoa Puff's trip to Apokolips where Khalid and Nabu become best friends and easy lovers.
Making friends the John Constantine way!
Doctor Cocoa Puffs continues to hamper Constantine's escape attempt. He can apparently shoot Ankh-shaped fish hooks into the portal that John opened but he can't follow himself unless John allows it. In his fishing attempts, he manages to snare Earth-2 John's Mom and pull her off of Constantine's Tenser's Floating Disc. Constantine's fake family get even angrier.
You know, John, some people's base motivation isn't simply self-preservation at any cost. I know! It's crazy. It should be but it isn't for some reason!
Constantine and his group (composed of many more people than just his pseudo-family and quasi-friends. I guess he just grabbed up everybody near the spell in London) travel through The Green and The Grey and The Red until they wind up in the Astral Realm. Even here, Parademons attack a blue flame that is the source of life in the universe. Is this like that light that Lobo kept trying to snuff out in the Magenta Timeline? But Constantine ignores it because fuck the Earth-2 universe! Put a spork in it, as the kids say, because it's too hard to eat with just a fork or a spoon. And then Constantine and his new frenemies are suddenly through into the Earth-Main-Earth universe! And they brought a gigantic hand with them!
Apparently it's just Darkseid's hand. And he's suddenly become interested in Constantine because Darkseid likes to find new Universes to consume and Constantine has shown the ability to cross from one to the next. Constantine throws a bunch of survivors at Darkseid's hand because fuck them. They would have already been dead anyway if it hadn't been for Constantine. Now he's trying to save Universe-Main-Universe and the only way to do it is to pull a con on Darkseid and he's only got one issue left.
Constantine #22 Rating: No change. Darkseid didn't recognize Earth-Main-Earth's universe but that was probably because only his hand made it through the portal and Darkseid is no Ten Eyed Man. If Darkseid had known which universe Constantine had escaped to, Darkseid might have retreated of his own free will. Not the universe that kicked his ass! Not the universe with The Batman that humiliated him and his son and his Glorious Godfrey! Not the universe that put Highfather in his place with a proper scolding from Hal Jordan! No reason to kick that universe full of hornets when there are 50 others to kick that are full of easier victims, like cartoon rabbits and Chibi Superheroes and Marvel families that can't even say "crap" or take advantage of Georgia Sivana's wonderful traits.
I just realized next month is going to be a pretty exciting time with Last Issues all over the place!