Marvel's Avengers won't support "Join in Progress" drop-in matchmaking or crossplay at launch.
In an overview of the co-op features posted on Square Enix's official website, it details that players won't be able join you if you're midway through a mission, but they will be able to jump in before you start one or after you've completed a mission in the lobby screen.
"With around 80+ War Zone/ Drop missions available at launch, you've got lots of options on what to play in Marvel's Avengers, and who to play it with!
You can: Enable matchmaking and jump into the game with other Marvel fans online, invite your friends and form your perfect Strike team, disable matchmaking to fly solo and take on War Zones with AI companions kitted with the gear and skills you've earned, or a combination of the above" a post on the site states."
The post also gives you a look at the "flow" of a mission, so you can get an idea of how matchmaking and co-op will work. Players will load into a lobby, and if you have matchmaking enabled, an online Strike Team will assemble. If you start a mission without filling up your team entirely, the remaining characters will be taken up by AI.
You can also manually invite your friends to join your team by "hovering over their Nameplate in the Social tab of the character menu." The Strike Teams will also share a unified power level, so, for example, if a buddy has a higher power level than you, it will balance it out across the entire team. Rewards also aren't shared, so anything you earn will be yours alone.
In the technical tips section, it also states that both join in progress and cross-platform play isn't supported "at this time", so it could very well be something the game will look to add in future. For the time being, though, both features won't be available when Marvel's Avengers officially launches on PS4, Xbox One, and PC on September 4.