I think what is happening at Dum is most likely to be a mix of different effects, and a plan long in the making as a contingency. If you break it down with the Dwarves having a never Chaos belief and experienced this for thousands of years, then you understand that the dangers they are facing comes from 1) Magic Saturation and The Environment, and 2) The Chaos Warriors themselves.
They have the Waystone that in the past may have sufficed to create a livable level of magic saturation, particularly if you limit your time spent outside but as the chaos wastes gradually expanded and the magic got more saturated this natural ability may have gotten overwelmed. Or perhaps more likely, the Dwarves themselves need to physically be outside for some certain period of time to collect resources but the dangers of that just gradually increased. So fundamentally they needed a better way to turn their environment towards their advantage, and to reduce the magic saturation levels.
The second problem is rather more obvious and one that the Dwarves as a race have been facing for millennia, that of more enemies constantly fighting them and wearing them down over time. With Chaos as a foe with it's wide influence and at times demons literally at your doorstep, this is all the more the case. So fundamentally they needed an armed barrier to be able to supplement their forces, if not manage it themselves to minimize the attrition to themselves.
So while the "what happened" may not necessarily be figured out, the "why" does as this seems to have fundamentally addressed both things, and given we know that the flow of energy down the Waystone remains pure they are very much still Dwarves and not just another version of Dhar Dwarves (not chaos ones obviously). They just may not be Dwarves as acceptable to the Karaz Ankor, or at least not acceptable enough to be willing to provide help, when they're in a situation that the environment itself could be killing them hence Borek's act. Prime example being Thorgrim's response when Mathilde asked for help during the Waaagh.
When you theorize it, and Mathilde should have all the knowledge needed, the Runecrafters/Runemasters of Karag Dum should realistically be the most skilled and potentially learned in the entire Karaz Ankor. We know from the Demon's blood experiment that Runes take energy from the winds to power themselves, so Runes that are placed in the north should understandably be more powerful or at least more consistent if there is more to draw on. This goes further if you consider runes that are no long effective further south and thus gradually over time were forgotten, would have still worked so far up north, and given what we know of Runecrafters and Dwarves in general they wouldn't really be inclined to share this knowledge. Then there's the casual knowledge of the Waystones that Borek displayed that is unknown to other kings, but it'd make sense for that to be more known here if it's not only core to the running of the Valaya runes themselves, but reducing the saturation to livable levels for them. Plus if you are aware of both the Waystones and the wider impact Runes can have on a large scale, there is obvious potential for further experimentation on other wide effects. Such as what is happening here, which is the beginning of the answer for "how" they are doing it if not an explanation of the mechanics of it.
Finally is the what, and given the two different effects in place you can take them separately given they serve different purposes. We already know the chaos wastes are very mutable and themselves are also capable of translocation (that soldiers childhood house = maybe Chekovs gun), so if you understand that they are capable of that naturally, you can likely direct towards things more beneficial for your purposes. We also know that translocation is possible from other areas such as the various magics that can do it, the Wood Elves with their world roots, but importantly for this the Beastmen themselves showing a similar capacity to travel between separate forests without crossing the land enabling them to crusade or "gather from all across the world" that has been mentioned in canon. With the Karag disguising itself as a Legendary Herdstone, as well as some manner of controlling Morghur himself, you now have a way to rally a force from the entire planet itself with which to defend you and a reason to do so given how their behavior around Herdstones have been described. Who knows how the dwarves could have theorized this one up honestly, but it wasn't shocking for Borek to see it, and I've got to say it undoubtedly works. Perhaps they had contact with one of the Wood Elf factions particularly as they also have their own Waystone network that powers their own stuff, and they sort of collaborated on taking out a respawnable enemy and threat. The Dwarves get a disposable ablative shield, the Wood Elves neuter their enemy and can claim more forest for themselves. This would also be another reason to keep it hush, given the various attitudes towards Elves. Though this doesn't really explain Morghur's affectionate attitude.
The changing of the surroundings into the desert is more simplified given on the one hand you've got Karag Dum in the chaos wastes constantly saturated in magic, and on the other hand an area from the equator furthest from the Polar Gates. If you can figure out a mechanism to get the translocation started, it may be easier to continue given the natural mutability of the chaos wastes particularly if mechanically "magical pressure" is a thing and it would prefer to stabilize itself and travel to a low energy place. This may even be an explanation for Mathilde on why and how the Winds blow the direction they do. Regardless, this is literally the Dwarven way of declaring a war on the chaos environment itself to get rid of it as well as to ensure it's more survivable for themselves. The sand itself isn't even necessarily permanent either, as over time it can be terraformed into fertile land particularly if you have a biome nearby such as that handy forest.
This would also more than explain the Yusak's disfavor with the Chaos God's if the Dwarves entire plan is self sustaining, for as long as Morghur remains within their control. The more magic in the air = the faster it grows, the less magic in the air = the slower it grows.
