Disclaimer: I know you asked for a way to gently kick someone out, but my answer is more about ways to let them play with you with reduced pain. My belief is that kicking someone out from your gaming group would damage your relationship, and there are other things to try before.

I was in a similar situation, playing a team online game with people that are so toxic even in written chat they eventually got temporarily banned. We had some heated discussion about the game objectives and decision made, and about behavior. I kept in good terms with most of them, and set a different path with another (we met in the game so we weren't close friends though), so I believe I can share a bit my experience regarding how to handle the situation.

If you want to play semi-competitively with a group of players, which I can imagine at least this player wants to do, you *should* setup rules for how to use the vocal chat:

> However if things start to look slightly bad during a game this player gets demoralised very quickly and starts criticising the other players more and more lowering the morale of the group and usually resulting in a loss. 

A fairly simple rule here, that wouldn't offend anyone: game time isn't criticism time. This is not only, from a casual gaming perspective, an undesirable thing for the whole morale and fun of the game, it is also from the perspective of a semi-competitive player an unnecessary distraction, which make it easy to agree for everyone.

You can, for the need of progression, and if players agree, have  **retrospective** criticism. Saving remarks for later make sure people are cooler, have access to replay if applicable, and can focus on the arguments instead of what's currently happening.

If you ask for this rule and you get players to agree, you would have grounding to tell live to the toxic player that he should keep his remarks for later.

If you have a loss where people want to retrospect, insist on it to be constructive criticism: blaming someone for the loss is not constructive. Loosing is part of the game, and focusing on that outcome has no improvement or team building value. Improvements opportunities are equal in won and lost games. Make sure these principles are well accepted especially by this player.

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This is how to handle it withing the group but a little head's up could also be necessary.

I know a few players that had personal sensitivity to anger and were not controlling themselves as well as most people do. You can first have a conversation about the possible anger issue. You could say "You seemed very angry last game. Anything wrong ? Are you still having fun playing, even when we loose ?". It's a bit fishing about whether or not they agree for an apology, and for the future, warn them (sometimes, with their agreement) about countermeasures. It could be the player confess something else, like they'd like to play with someone else or to another game.

It's also about making sure they understand there is a problem, and that the subtext is that it's not really acceptable, even if that is generally useless to say. Remind them, that it's just a game, and you expect it to be anger-free. You could also say, if things are not going too well you keep yourself the freedom to cut their microphone (if you can cut for everyone, that's best)

Once you had that discussion, even if that seem a bit extreme, you can drop in the vocal "Sorry but I mute you for the reminder of the game" with several desirable effects : you echo to the head's up discussion, you get a game free of anger, and you dissuade from recidive. No need to add on the incident: most people would get the message fairly well.

If everything fails, you could consider playing another game with the toxic player, possibly without competitive aspect. Sometimes, it's just the game just doesn't fit the personality. You could consider this as a last resort from effectively kicking him out of your gaming sessions.