There are two kind of battles, separated on four difficulties.
The first two battles, on Beginner and Intermediate difficulty, take place on a random Story map with three random enemy heroes. On Normal difficulties, 3 foes at the level of your strongest hero minus 5 await you. On Hard difficulty, they are 4, with a difference of 2 level. Clearing these battles will grant Memento Point, based on the following formula:
Where is a random multiplier dependent on the enemies drop as describe below:
1 drop: 1.0 / 1.1 / 1.2 / 1.4
2 drops: 2.0 / 2.2 / 2.4
3 drops: 3.0 / 3.3 / 3.5
The Boss Battle and Final Battle are unlocked after reaching respectively a Rapport of 6 and 20. They feature a random Heroic Ordeals from Book 2 or Book 3 and above, depending on the difficulty. These battle do not reward Memento Points, instead their first clear grant unique reward, such as Divine Code: Ephemera and accessory.
Once acquiring at least 100 Memento Points, a duo of heroes can take part of a Memento Event. Each event offer 5 different Memento Event, with three possible outputs. Based on the result of the Memento Event, the active duo will earn an equivalent amount of Rapport.
Memento Points are capped to 9,999, while Rapport are capped to 99 for a given pair. Changing one or both member of the team will reset the Rapport counter to 0, event if the same pair is rebuild after. Changing for another version of the same hero will also reset the counter, however simply changing the order of the currently selected heroes won't.