Diablo – What triggers a follower’s armor progression

There isn't a way to manually trigger followers' abilities.

According to Blizzard's Follower FAQ (archived here):

Will you be able to control the followers?

No, we've avoided pet and follower commands as they added unnecessary micromanagement to the game.

Sadly, no, there does not appear to be a way to check your follower's cooldown timers.

As far as I know, the portrait and the follower panel are the only methods of getting data about your follower's current status.

I tested this with my Templar, by giving him a bunch of active skills and then putting him in situations where he'd want to use them. His portrait doesn't change at all, regardless of his ability states, and there's no indication on the follower panel that any of his skills have been used.