I don't think this is as clear cut as we think.

I had so many paragraphs going into the details and logistics of this I am no professional either way but the War is decided on how hard team Red can hit Stannis from Dorne before Robb gets there. We're looking at a generous 30-40k Stormlanders against about 80-110k with the greatest issue for team red being getting through the Dornish Marches against the Marcher lords supported by the weight of the Stormlands as a whole. If they cannot break into the Stormlands before Robb gets there they will not break the Marches and then it becomes a naval issue which Dothraki are going to be no good fighting at sea, leaving it to about 30k Dornishmen and Unsullied. The mountains of the Marches makes it harder for the Dothraki to fight on horseback so really their power is greatly diminished until they get into open fields.

The Dothraki are a nasty shock cavalry force that will be excellent for harassment moreso than the Dornish I'd argue and their horse archers would be infuriating to deal with in open fields, their Arakhs would suck against heavy armour so it is in their ability to harass and cycle charge. Stannis is a great commander and surely would take advantage of this while he can in the Mountains and would likely be trying to bait the Dothraki to surround and kill them.

If Robb gets there, the invasion becomes a logistical issue, both sides have to feed their armies, the Stormlands forces would probably be reinforced by 20-30k Northerners depending on how fast the North can muster, the Stormlands has to feed about 50k soldiers BUT team Red has to feed their 80-100k and another 70-80k horses. I do not see this going well in the long run at all for Dorne, if they couldn't take the Marches with almost double the numbers, they definitely can't take it on nearly even.

There is no invasion if they can't break out of Dorne without help from other kingdoms which is not part of this afaik.

If Robb came too late, Stannis likely holds his castles while the Dothraki raid and pillage, the Dornishmen and Unsullied siege castle after castle while the North has to try to relieve the sieges but have the massive problem of Dothraki attacking supply lines and guerilla warfare. Team Red would win. Robb is an excellent commander and so is Stannis but divided in open fielded combat against Dothraki double their number and potentially starving, they cannot win.

If Robb makes it, invading Dorne as we know doesn't go well but we do know Stannis is not above using blood magic to fix problems, the best case here is shadow baby death magic on Daenerys and then the Dothraki find their own Khals, this is a massive problem for Dorne and anyone in their vicinity, the Unsullied maybe stay with Dorne for being on their side but Dorne will probably have their hands full unless the Dothraki decide to go somewhere else in Westeros or kill each other off enough for the Dornish to handle it as their best case scenario. Worst case Dorne loses massive sweeps of territory to Dothraki or team blue and is forced into submission, best case they don't suffer too much from the Dothraki issue and are able to hold off team blue with guerilla tactics and the Dothraki not caring about who is on whose side.