I’ve ridden the trails around my place for the past 13 years with herding mutts. The very best was a collie mix, followed by border collie mix. Currently I ride with my mostly GSD, a BC/retriever, and my designer dog “Texas Heeler” ( half Blue Heeler, half Aussie Shepherd.)
With the exception of the collie mix, all were taken for their first ride as soon as they reached the age of reason, roughly 1 year. They all were schooled on the trails as pups both on and off leash and 1st rides were short and close to home. I don’t cross roads with them. I’ve got access to a couple square miles I’ve ridden with them.
The collie mix very reliably trotted along right under my stirrup. The others are mostly within sight but not always. The Heeler thinks it’s OK to grab an uncounted horse by the tail but has not tried this stunt on a mounted horse. All 3 of my trail dogs have radio collars, handy things that can help you locate the dog by sound and reach out and touch them without dismounting.
I have also been on an all day ride in a big group that was escorted by an Aussie Shepherd, and joined on dirt road rides for miles at a time by hounds and a pitbull who decided they’d rather join a parade than watch it. Also by my late mother’s 16 year old generic mutt for short distances.
Dogs wear orange vests in hunting season. So do I. And horses I ride have orange tack/saddle accessories. As regards heat tolerance, the Heeler was bred for endurance driving cattle across the Outback, the Aussie Shepherd for herding sheep in the arid American West. All my dogs can and do avail themselves of ponds and streams to cool down and stay hydrated. But I don’t take them out as much in midsummer here in GA as I do in the cooler seasons.
I’ve also owned lab/spaniel mutts that were NEVER off leash except with direct and close supervision because they would chase deer into the middle of next week. When the herding dogs jump a deer they give it a brief escort and then come back to the horse.