I run a DC Adventures variant game in a distant space sector. The sector's Green Lantern (a legacy character) has gone missing. His ring returned, damaged, to Oa and the Guardians have been unable to discover his location and believe he may have been killed, fallen out of this dimension or has shifted in time. They assigned a new GL to the sector, and this is the player's character. He constantly has to deal with the old GL's legacy and how he isn't quite up to snuff ... and may never be.
In one segment, I had the legacy characters (I call them the World Guard, or High Guard) were dealing with the attack of an interstellar dignitaries ship in orbit. The player's group was able to pick it up on the monitors and watch part of the action. As things quickly progressed, the attacked ship released life pods which fell to the planet's surface. The High Guard sent out a call for on planet heroes and emergency services to respond to the landing sites and help surviviors. This became the core of that game session, as the players discovered that their pod had the ambassador in it & was being trailed by an attack fighter. The pod also had a traitor aboard and some of the aliens were super powered in their own right.
The point being, depending on your campaign, there's a lot going on and everybody isn't going to get involved in every adventure. You can even use the other character's as background noise to add flavor to a game or seed plots for your players.
If the big brawl knocks down a building ... who helps survivors and stumbles on a gang of bank robbing Morloks?
The daily news feeds could reveal that the Freedom League is dealing with a coastal Tsunami or strange lights seen on the Moon and that Police predict an uptic in illegal activity while the group is out of town.
One or more of the League goes missing after a particularly odd encounter. While the rest of them look for their friends, the players stumble across indications that at least one of them has been captured by a villain and realize they have to act before its to later.
A Panic Plague hits the city. While the Freedom League deals with the fallout, the party discovers that one of the inmates in lock up is actually the problem. They then discover that lock up is under the control of a new (or existing) foe and have to fight or trick their way into the facility in order to stop him or her.
There's any number of options depending on how you want to play it. Remember, some comics have big events that never see a different group or hero. It's just assumed they're off doing something. They also have personal lives that may make them unavailable ... for whatever reason. Sometimes, stuff happens and not everybody can be around to save the day.
Armor goes off line due to damage or malfunction. Heroes travel to distant homelands (you could even have one show up & give the heroes a 'heads up' that he or she will be gone). Powers go wonky or fail, taking heroes out of action for a time. They have to deal with a family crisis. Lots of stuff. But, you don't have to script it all. Do what your plot and story call for. Let the rest go unless you want to use it for something.