It could vary a lot. Maybe they get into a dangerous adventure and face some unexpected enemy attacks.
Well, it might involve a complex plot. Maybe they're on a mission to save a planet and get injured in the process. Or it could be a trap set by a villain they underestimated.
One idea could be that the OC is a time - traveling journalist who meets the 10th Doctor during his adventures. They initially clash because the OC is always trying to get a good story, but eventually they become friends and the OC helps the Doctor solve some mysteries. For example, in a planet where the inhabitants are losing their memories, the OC's sharp observational skills come in handy.
In this fanfiction, the 12th Doctor might face all kinds of adventures and challenges while being kidnapped. It could involve his companions trying to rescue him and uncovering a big conspiracy.
It could lead to a wild adventure as his companions race to save him. Maybe there's a villain with a specific motive behind the kidnapping.
The story might involve a complex plot where the kidnappers have some sinister motive, and the Doctor has to use his wits to escape.
It could lead to all sorts of adventures and unexpected plot twists. Maybe there's confusion at first but then they team up to save the universe!
The story might have the 10th Doctor rush to the reader's aid, using his medical expertise and Time Lord abilities to try and heal the injury.
In the fanfiction, the 11th doctor, with his eccentric and energetic personality, could initially be a bit confused by the 13th doctor's more modern approach. But then they could start to combine their wits. For example, the 11th doctor's knowledge of ancient alien races could be paired with the 13th doctor's understanding of current galactic politics. They could work together to save a planet that is on the brink of destruction due to a complex web of time - related issues. The encounter could also lead to some humorous moments, like when they argue about who has the better sonic screwdriver or who has had the more exciting adventures.
In some fanfictions, the 10th and 11th Doctor are brought together by a time anomaly. They are forced to cooperate to fix it, and along the way, they encounter old friends and enemies. The contrast between the 10th's more traditional style and the 11th's more zany approach creates interesting dynamics. They might also have to deal with the consequences of their previous actions in different timelines, which adds an extra layer of complexity to the story.
They could also meet in a time - locked situation. The 12th Doctor is trying to solve a mystery from his past, and the 13th Doctor stumbles upon the same situation in her present. At first, they don't understand how they are both involved, but as they start to piece things together, they discover that it's all part of a much larger, more sinister plot. Along the way, they might meet some old friends or foes from their respective tenures as the Doctor. And in the end, they work together to outwit the mastermind behind the plot, all while having some really interesting conversations about their experiences as the Doctor.
They might have a friendly competition. For example, who can solve a time - related mystery faster.