Are the mounting posts and position for road disc brake calipers the
same as or similar enough to mountain bike disc calipers that you
could put a set of road disc brakes on a mountain bike or vice-versa?
IS and post mount calipers are cross-compatible and can be seen on both MTBs and road bikes. Flat mount clipers, on the other hand, are a new thing that was designed specifically for road-type usage, with very limited presence on MTB frames (some small builders and Cannondale bikes). You can put a post mount caliper on a flat mount frame/fork using an adapter, similar to IS-to-post-mount.
However, my question here is narrowly about how you would convert the
brakes of a disc-equipped mountain bike to something drop bar
Certainly possible, but details depend on what calipers MTB has. Shimano hydraulic calipers are compatible between road and MTB levers. In fact, BR-R785 was an XT. I don't see why you can't just get a set of any Shimano hydraulic road levers and put them on. No idea about SRAM, though. You should also be aware that mechanical MTB/road calipers might have different leverage ratio, like BB7 does.
If that can be done, what are the downsides to be aware of?
Except for existing parts compatibility I see no reason why anything might cause trouble. I'd be concerned about geometry though, drop bar will increase the reach significally, you might want a shorter stem.