I sometimes ride with our dog, an almost-blind Shih Zhu (genetic issue rather than being particularly old), in a backpack. It's designed for dogs, so it has a hard base, mesh side panels, and a clip for his harness. He can curl down and sleep in it, or as I'm leaning forward whilst riding he can lie up my back, with his head over my shoulder.
I call it my babe-magnet bag, as it gets me lots of attention from ladies young and old (dog is a bit of a cutie, looks just like a teddy bear) - to MrsH's amusement as she's normally with me if I'm riding with the dog.