(OOC: The rules of thumb from Sandstorm (not RAW in LEW because they aren't in the SRD, but definitely a good idea to reference) help flesh out the different desert hazards, which also include sunburn, sunstroke, and heat exhaustion. They have a chart there too, and I believe that any two 'Desert Born' + 'Desert Wear' + 'This Salve Thing, can't remember the name, but it works like Lasair's' will keep you safe in standard desert. Extra harsh desert (like the one in the land of fire) requires all three, which is why Lasair's people have all three. Unnaturally super hot deserts (also present in the LoF) will require saves even with all three. Fortunately, Endure Elements counts as something like +4 on the scaling (each of the aforementioned counts as only +1) and keeps you safe until you get to magically crazy hot deserts. For all of these values, the hottest part of the day usually counts as one step up from that (it's based on temperature numbers), which is why many people will pitch tents and rest for that time).)
So in other words, if we follow those rules of thumb, which are good common sense, Shadya and Lasair will be fine in normal travel. If we don't, then pretty much all desert-dwelling civilisations would fall apart thanks to the death of their commoners.
Rae, do you have Sandstorm? If not, we can sort of just handwave the specifics for everyone except Cludge, I suspect, since we have desert dwellers and Endure Elements, and he seems the only one who will try it with neither (and with heavy armour).
Our movement speed has actually not changed, though we will probably rest during the hottest parts of the day and thus do something like travel four hours from dawn, stop for the hottest six hours, then travel four hours again)