There's no consensus etiquette. It's whatever suits you. It is your prerogative to expel anyone who is timed out - the button is there, so you are allowed to click it.
But, I have been grappling with how to handle time-outers myself ever since I started playing here. It is odd that the site lays the decision (and thus the responsibility) in the hands of the other players on what to do when someone times out. I don't think I've ever seen a similar system in any other game or gaming site. Usually, if a user breaches the time limit, they automatically lose / quits the match. But, the BGA system is what it is.
Here's my personal lines of thinking, currently.
In Arena real-time: I kick immediately. Arena is competitive. Clock management is part of the skills we're competing about.
Regular real-time: I ask in chat. If they don't respond for a minute or two, probably expel.
I have tried the "turn into turn-based" option once because that seemed like a promising solution too, but IIRC it automatically set the speed to 1 move / day then? That is too slow for me in the majority of Turn-based games. If the feature allowed me (the one not run out of time) to define the speed of the turn-based game, I would use that for many games I think. So rather than sit staring at a screen waiting for someone who has probably disconnected, or turning the game into an excruciatingly slow turn-based affair, I unfortunately have to expel most of the time.
Turn-based: I have just now started trying to reach out via direct message to players who are out of time, or about to run out, with a very friendly reminder. I suspect some people don't notice what the speed of the game is set to. Others, I'm thinking, may get a reaction from being reminded that oh, there are actual real people waiting for me over here so maybe I should not ignore the clock.
So far this method has helped for 1 player out of 3 attempts.

If they respond and explain, like "Oh sorry, I forgot, I'll take my turn tonight/tomorrow" or whatever, I don't expel of course. Anyone can have a busy day or forget.
If they don't respond at all, or continue to go out of time regardless... Well, here's where it becomes tricky. What I want to do factors in by:
- Is this a game that is very hard to find players for? If it takes a week before I can find another table, it's a bummer to have to end the game.
- Is the game just beginning or soon about to end? If a player immediately shows problems with keeping time from the start, I'm more inclined to expel right there and then and try again with other players. If there's just one turn left and the player played reasonably before, I'm much more inclined to give the benefit of the doubt that they will come back soon and finish the game.
- In a multiplayer game, what do the others prefer?
... But safe to say, if they haven't been heard from in
six days, I expel them.
Tournaments: I don't play tournaments, but there are reasons to
not expel anyone in tournaments, at least not in multiplayer elimination. As a new user, you will end up eliminating
yourself from the tournament. This is because the
newest account gets eliminated when there is a perfect tie (and the system considers an expulsion to be a perfectly tied win between all non-expelled players). But of course you can't control whether someone else chooses to expel, so you can arbitrarily get eliminated due to someone else running out of time.