I was laying awake last night thinking about how I would convert the Perry Agincourt into Dornishmen, as you do, and I figured out how I am going to do things when I get around to the Dornish. I don't really trust other manufacturers to scale well with them, as the plastics are slight for 28mm, so what I will do is sculpt on head scarves and clothes.
I don't think that the Dornish line troops would wear as much heavy quilted gambesons as the rest of the Westerosi would as they would be unbearably hot in the Dornish sun, so I would either fill in or file down the lines of some of the quilting and lengthen the skirts to make them look more like desert clothes or sculpts cloth skirts beneath the armor. I'm not sure that Sand Dornish would really wear turbans, but they would wrap their helmets and armor in cloth to reduce the metal's exposure to the sun, much like the the Saracens would do.
As for the men at arms and knights, I think that the French would be better suited for them for all but the Stone Dornish in the north. This is not for the same reason that they would be better suited acting as Reachmen or Stormlords, who come across as kind of poncy stereotypical knights to whom heraldry and looking good is as just as important as fighting. They would be better suited because the French knights are wearing clothing to cover the metal, which would reduce the heat of the armor. I would also sculpt head scarves on many of the helmets. For the Stone Dornish of the Boneway, I see them are far more martial and serious about warfare than many of their southern brethren and would probably wear more white armor like the rest of Westeros as they do not have to worry about the heat as much.
However, I do not think that the Dornish will field nearly as many knights as the other parts of Westeros, and would have only one or two units of them in an army, with the rest of the army being made up of light infantry (spears), light cav (sergeants, bowmen, and javelins), and skirmishers (archers or javelinmen). If I were to do a Lion Rampant list for them, these would be it:
Stone Dornish: 24pts
Mounted MAA-6
Mounted Sergeants w/bow-5
Foot Sergeants-4
Foot Yeomen w/ javs-4
Archers-4
Sand Dornish: 24pts
Mounted Sergeants-4
Mounted Yeomen-4
Mounted Yeomen w/ javs-3
Foot Yeomen-3
Foot Yeomen-3
Foot Yeomen-3
Archers-4
I feel that, since Dorne is one of the poorer parts of Westeros, the English longbow bodies would be a little more common, as they have less armor. This would also allow for more static poses for the stalwart Dornish spearmen, which they are famous for.