Does the intensity of the Reach/Dorne rivalry vary throughout each kingdom? I would imagine the border lords have the most hatred, but then again, Arys Oakheart talked about his family killing Dornishmen even though they’re not anywhere near the Marches. So what about the Redwynes or the Florents or Rowans, etc?

That’s an interesting question – certainly I would imagine the marcher lords care most of all, since they’re the ones who have to deal with raiding on a regular basis. And I would imagine that distance plays a factor – so the Rowans, Cranes, Fossoways, and the other lords of the northern plains would be more focused on the Westerlands or the Stormlands. 

But as you said wrt to the Oakhearts, given the frequent wars between the Reach and Dorne, and indeed between the Stormlands and Dorne, I imagine there’s a good number of houses who have fought enough wars to have a personal stake in the matter. Entire armies don’t vanish in the deserts without thousands of families being affected.