Thoughts on the hypothetical concept of flying mounts? What would it take for a flying animal to be both light enough to fly and big enough to carry a human, along with being domesticated like horses?
Hmm. The thing is that large flying animals are strongly selected for being as light as they can possibly be -- birds gave up teeth and external genitalia, turned their bones to sponge, and developed special ultra-dense cortex so they could miniaturize their brain -- so I don't expect there to be much margin for cargo or riders. Any weight they can carry, they will occupy by themselves.
Your best bet is to have a flyer so large that the weight of a human is relatively negligible. The largest flying known to ever exist are Azhdarchid pterosaurs like Quetzalcoatlus, Hatzegopteryx, and Arambourgiania; most estimates seem around 200-250 kg (some get as high as 400-500 kg, but at least one of those claims flightlessness), which perhaps would allow one to carry a child or an especially light adult. But you'd have to be careful not to mess up weight distribution and aerodynamics.
I'm not sure heavier-than-air flyers larger than Azhdarchids are feasible on Earth (then again, if we'd never found their fossils, I'd probably say than flyers heavier than Argentavis aren't possible...) Blimp-like lighter-than-air flyers, with bags of biogenic hydrogen or methane, are theoretically possible, but AFAIK none has ever existed on Earth. The issue with biological blimps is that (since balloon membrane weight increases with the square of its size, and buoyancy with the cube) small balloons are very inefficient. See the Furaha blog for more. The way I solved that issue in my own work was to have methane bladders develop first as metabolic reservoirs, and only later be exapted for floating.
