I saw one crossover once where the Doctor landed in Panem and just left. That didn't quite make sense to me, so I decided to explore a little further. Basically, he accidentally lands in Cinna's appartments one day...

Cinna has read of this man in the old records, a man in a blue box who saves people. But he never expected to see him in person. And he never expected him to refuse.

"No. I can't stop these Games." The man looks away from the screen as one Tribute stabs another in the throat. "I didn't even mean to be here."

"But now that you're here, do something. Stop this madness. I know you can." Cinna pleads. "Seventy-three times, twenty-four kids have walked into that Arena. Only one walks out. Only one."

"Yes, I could save them. I could take every single one of them out of this death trap. Even that girl choking on her own blood—I could treat her, bring her home on a hospital bed instead of a coffin. I could blow this whole arena sky-high."

"Then why don't you?"

"And then what? They'd all disappear quietly. A robbery gone wrong, a hunting accident, food poisoning, an infectious disease. And the next year they'd begin the whole process over again, only worse. You know it would."

"Then go after the government. You took down one with six words once."

He stares vacantly. "Have I?"

Cinna stares at the man's jacket. Leather, he thinks, but not the artificial variety that was popular a few years back.

"You say I should take down the government. I could find each of them, shoot them in the head or shove them into a supernova. And maybe for a time it would stop. But after a while—a century, a decade, a year—it would start again. People would surrender their freedom for security, their rights for entertainment. And in the end they'd curse and beg and pled, but as long as it's somebody else's son, somebody else's sister, somebody else's sibling—as long as it isn't them, they'll submit." The Doctor stands and walks back to the TARDIS. "Start your own revolution, Cinna. That's the only way it will last."