Slitherine

Do military buildings like training ground, archery, barracks have any purpose if they are not in a provincial capital? Do they add anything when units are raised by an raid attack, the urban militia and such which disappear after the attack? It sounds like they would from the description of the building but it isn't clear.
I read that some of this may be changed in future patches. What should it be, just by the logic of the game?

Currently, the main reason for building XP boosting structures outside of a capital is to unlock higher tier Military structures, most notably bigger Walls. A cheap Training Ground with no slots is perfect for this.

But as you say, changes are likely in coming updates ...

to your basic question - no they have no impact on the auto-garrison, but some of the defensive buildings like the siege workshop do.

its possible that a future patch will change the current system so that these buildings generate +xp for new units regardless of where they are in the province (there are also a few non-military buildings like the advanced blacksmith and hunting lodge that do this too). I'm not sure about the gain to this, not least its going to make it harder to be sure that you (a) have one of each and (b) don't overbuild whereas at the moment you just need to keep an eye on your provincial capital.