[Western Crossover] Did the Doctor Pull the Handbrake Today?

A note before reading: The protagonist is not the TARDIS personified, and the protagonist has no CP throughout. This is mainly about a parent taking their kid traveling. There are ship hints for the Tenth Doctor and Rose Tyler, and Eleven and Song Jiang. The official show is better at this than I am, after all. (doge)

Years later, the Type 40 TARDIS looks at another of her kind speeding past next door and shrieks at her thief: "Look what you've done! Another Time Lord who forgot to pull the handbrake!"

The Doctor, entering coordinates, opens his eyes wide and looks at his Rose with the wounded expression of a falsely accused puppy. "I really did put the handbrake on this time."

Rose leans against the railing, laughing so hard she nearly doubles over. "Yes, darling, but this time you floored the accelerator. We've reached the end of time."

This is such a niche interest that I'm writing it for myself. There probably won't be many readers, and it likely won't be contracted, so I'll just cook at my own pace and entertain myself.

After all, there's already one last pepper pot. What's wrong with giving the Doctor one more companion and a home? I'm completely justified.

Self-indulgent ships included: Ten/Rose, Eleven/River

Shows covered so far (the parts in parentheses are relevant to the main plot):

Criminal Minds (Season 1 onward), Supernatural, Good Omens, Transformers, Marvel, DC, and other Western classics

Theme: Life isn't just about stepping on the accelerator. Sometimes you have to use the handbrake too.