Out of the 3 battlegrounds I've tried to play:

The first one kicked me for being 'inactive' (as described above), not letting me do anything. I got hit with the deserter penalty.

The second one was capture the flag, which was not fun to play the first time around.

The third one hung on the loading screen (I heard the load-in sound, but the loading screen never went away). I restarted ESO, and by the time I got back in, the game had decided I was gone, kicked me and given me the deserter penalty. I believe my team did not even get a replacement here, the game just automatically kicked me.

At least for the third one: don't kick the inactive/offline player unless you have a system for putting in a replacement AND a replacement has been found and is queued into the game. This should be obvious: if the offline player loads back in in the middle of the match, they should not get the deserter penalty. If they don't load back in in time, then slap them with the deserter penalty (reduce it to say 10-15 minutes, start it after the match ends).