One possible plotline is that they meet in a time - locked situation. They have to work together to break the time lock. Another could be that they meet because of a shared enemy that has somehow found a way to target both of their timelines. They would then have to put aside their differences and use their individual strengths to defeat the enemy.
In fanfiction, when the 11th Doctor meets the 12th Doctor, there could be a clash of personalities at first. The 11th is so full of child - like wonder and the 12th is a bit more reserved. But then they might find common ground in their love for the TARDIS and for saving the universe. They could end up learning from each other. For example, the 11th Doctor might teach the 12th how to be more carefree sometimes, while the 12th Doctor could show the 11th how to be more strategic in dangerous situations.
Well, it varies in different fanfics. The 11th Doctor could be hurt by powerful villains, or maybe in trying to save someone. It could also be from internal conflicts or emotional stress.
It could be due to various imaginative plot devices created by the fanfiction author. Maybe in a battle with a powerful villain or during a risky adventure.
The story could involve the OC having a special ability or background that complements or contrasts with the 10th Doctor. Maybe it brings a fresh perspective to familiar Doctor Who themes.
Whump fanfiction about the 13th Doctor often explores the more vulnerable and difficult situations the character gets into. It can show her facing powerful enemies alone, or dealing with personal traumas. For example, she might be injured in a battle against a new alien threat and have to struggle to survive while also protecting her companions.