Disclaimer: Diana Wynne Jones is the creator of Dogsbody. I intend no harm to her or to copyright holders with this fiction.
A dog will follow its master to anywhere at all. Bright, obedient love with a knowing nose attached - you can hardly throw a dog off, even when you try.
In contrast, you'll find that a cat cannot be shifted from its home, because a cat knows perfectly well what it owns. It is entitled to the comforts of familiarity and independence.
Kathleen knows this, so the third time that Tibbles shows up at Miss Smith's doorstep, mewing imperiously, Kathleen looks at Miss Smith. She smiles broadly when Miss Smith nods, and this time Tibbles isn't taken back to the Duffields'.