When thinking about the ankle support provided by mid-height boots, it's commonly assumed that the height can help stabilize the ankle. But the height of a boot is only part of that story. The midsole construction of the boot itself is really where "stability" lies. You can still roll an ankle in a boot if the midsole under your feet is supple, or you hit a big enough rock. There is some stability provided in the height, but really the height of a boot can help extend the waterproofing coverage or work as a kind of gaiter to keep debris out of your boot.
To that end, can you run in a mid-height boot? You could. Merrel, Hoka, and Salomon have all transformed versions of their trail runners into mid-height boots to allow hikers the running shoe feel in a boot height.
But should you run in a boot? Personally, I'd say no. In techical trail running especially, quick turnover and precise steps are the key to avoid falls and increase speed, and for me, it would be difficult to do that with the reduced flexibility in my foot and ankle from the boot. I'd also be concerned with friction caused by the taller height of the boot, particularly around the ankle bone. It's not impossible to run in a boot, and you might find it works for you, but I'd tread carefully on your first outings to see if it's right for you.
If you're concerned with rolling an ankle on runs, I'd recommend using a stiffer running shoe, or practicing trail run techniques that focus on quick foot turnover first.
Hope this helps!