Several members here are from Eastern Oregon, there are at least 3 on working ranches. It depends on the situation whether cattle are worked using horses. ATV's are used more than horses. It is more economical and efficient.
Herd dogs are common and seem to enjoy riding on the back of the ATV out to range, rather than trotting along with a horse mounted rancher for miles before roundup begins. Dogs enjoy the ride to work. I bet the pups have heard Grandpa Dogs stories, a hundred times, how he trotted 20 miles up a steep mountain to the range, alongside a cowboy mounted horse, before his work actually began, dashing back and forth, trying to keep the cattle together and headed the right direction.
The old ways are seen more in South Baker and Malheur Counties.
@webfoot attends several branding a year that use rope and drag. I am not sure if the old campfire hot iron is used since the gas heated ones are more efficient.
Years ago, on a drive on the back roads connecting Arlington to Michell and along the John Day river and breaks, I was amazed how many old corrals had been upgraded with squeeze chutes and calf tables. I think the upgrades are due to not only efficiency and economics, but the fact that real skilled "cowboys" are few and far between. Contract cowboys are becoming a thing of the past.