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Screens/Events/EX Battle
When you first open the EX battle screen, you'll be greeted with a bunch of series and a button on the right.
Select the series of battles you want to fight (which featured bot you want to face) and then select the "Challenge" (挑戦) button to open it up.
You can find a comprehensive list of the challenges and their battle conditions on the Super Invasion page.
On the bottom left, these is the banner and full-clear rewards; your "Completion Rewards" (コンプリート報酬). In this image, it is a Cross Messiah banner, 10 Rubies and 10 Cyplasium.
In each panel on the right, you can tap of "Special Rules" (特別ルール) to see the battle conditions. Refer to the Super Invasion page for individual translations.
For the most part, all EX Battles have the condition of killing ALL enemy bots to clear the battle as a common condition in addition to the special rules; match mutators.
After selecting a mission, you'll be brought to the mission screen. Here are the details of the match; standard stuff (tap on the icon to see bot details).
In the middle row, there are your First Clear Rewards (初回クリア報酬) and on the right, your random clear rewards (the usual junk).
The bottom allows you to start the battle, for 60 energy. After that, standard fare with the bot selection screen.