I would base a crossover between the two universes on one all powerful being. It would start off with the most popular Abstract's of both universes being telepathically contacted by someone, or something above even their immense powers. And asking after what felt like an assault, who this being is, and what it is. It would periodically answer throughout the 12 part series in riddles of exactly what it was.
At first it would say that "I AM THE VOICE CRYING IN THE WILDERNESS". With nearly the force of creation Which would be so powerful that it manages to toss lesser Abstracts like Galactus, and Eclipso light years from where the question is asked ("who or what are you?"😉. So powerful is the voice, that it opens a rift that spans the length of both the DC, and Marvel universes in all realities, across all time, and space.
It's entire purpose is the end of all things, a decree is laid out; that the old would be replaced with the new. Death, and it's Avatars are given more power over life, and it's Champions. Eternity/Infinity begin to lose power, and the balance begins to shift.
The being unknown to all fictional characters in the story at that moment in time, eventually reveals itself as the the Ancient of Days. In a last ditch effort the cosmic Abstracts from both universes come together in counsel, and decide to launch an assault of their own against this being known only at that time as "THE VOICE CRYING IN THE WILDERNESS".
The agreement is made, and through a concerted effort, the Abstracts rip a hole in reality, and launch their assault. Flying through the illuminated rift, the battle begins, and for an instant they appear to be able to take back control of reality, but then are soundly defeated, and tossed out of the rift. They are defeated to the point of destruction, and then an instant later are remade whole, and the rift closes.
The abstracts no longer belong to one reality, but instead, when they were brought back from destruction after the battle, they realize that both universes have become one. As they were recreated, something new has been instilled in them, and war erupts between the Abstracts. All but the most powerful of them battle destroying huge pockets of space until the Living Tribunal separates them, The Spectre like Eternity is nearly helpless due to the nature of the Ancient of Days who actually happens to be God. They go back into counsel, and decide that the fate of reality must not rest upon them, but upon both universes champions.
Celestial, and Angelic envoys in the forms of humans appear before every hero in the universe, and tells them that it is up to them to decide the fate of reality. The Heroes go the JLA vs Avengers sort of route for a while, in order to determine who will be the main team to combat the forces of Death. So they battle it out to pick the cream of the crop.
That's what my premise would be, if I made a crossover like JLA vs Avengers, and it would have to be sketched by an awesome artist on the level of Jim Lee, or Dale Keown. Ron Lim would do as well.