What is the appropriate verb that I should use to fill in the blank below?
People march on parades, but models __ on the catwalk.
The models walk on the catwalk. Unsurprisingly.
Indeed, from that, one instance of a model presenting an outfit is called a walk as well.
You could use the word sashay.
walk in an ostentatious yet casual manner, typically with exaggerated movements of the hips and shoulders. (Oxford University Press, online)
"The model sashayed down the runway." (Merriam-Webster, online)
I think it can serve in a figurative sense, so they don't have to be walking in any particular way. Analogously, people marching in a parade aren't always marching in the more conventional literal sense (as in military or a marching band), but may just be walking.
You can also say to model on a catwalk.
I think strut is actually a really good word. But so are pose and sell. They sell themselves and the outfits. They are in a continuous pose knowing that the cameras are going non-stop.
If you want to be elegant about it, I'd suggest "promenade".