No. Only in Capture The King.
[Survey] Teams king capture rules

Yes, Yes
my reasoning is simple: a check is much slower than a king capture, you're only threatening to capture the king on your next move. But if Green eats Red's king, Green was faster, and Red has no right to make another move, he's dead (all red pieces are dead after green's move, so green is no longer in check (or pinned). green succesfully got out of check (or unpinned himself). So it should be legal.
I think (well, who am I kidding, I didn´t think at all) in FFA can capture the king, in teams has to respond to the check otherwise.

I mean I think both should not be allowed. In regular chess you must respond to check and a pinned piece to the king can not move, why should it be different in 4pc?
for me this questions is similar to
in both black cannot checkmate white
in 4pc chess when players turn, priority for own king looks fairly
with using logic , that when on your turn check is not a check - also mate is not a mate:
blue on his turn checkmated but b/g team as nothing had happened takes both knights
in this case game is over only if some king is captured, i think this idea is useful to be type of 4pc variant, but not as standart teams 4pc
Kings capture is main difference from normal 2pc , which i consider quite reasonable and similize to mate threat. It is possible only in situation with "hidden" check from right side player, when between the attacking piece and own king placed left side player piece. In this case looks logical that team has option to create counterthreats for opponents kings and do not give a time for king capture
example
green Qf1 is checkmate on red's turn and game over , no matter what happens with blue king in future game, otherwise the game should not end after elimination one player and will continue until the last survivor
also on last diagram if yellow give check to green, (ex. yllow bishop was on g11) , green didnt have time for checkmate or king capture red
pin example
yellow's turn and now rook is pinned and no matter that red can capture blue king and "making dead" blue pieces. if allow on player's turn to leave his king under attack, again we return to situation when mate is not a mate and only king capture ends the game
so in my opinion - answer is clear no/no on both questions

@valger2 in the above chess positions, black cannot checkmate white because:
U CANNOT HANG UR KING WHILE CHECK/MATING THE OPPONENT
by law of move order, ur king will be captured first. This is the very reason why the above kings should 100% be captured. They are hanging their king while check/mating their opponent.
ok, to better visualize with the first post, find difference
greens king checkmated on his turn, but they win a game? didnt know any chess variant, where the game continues after the mate on the board
i understand what you mean, king capture > check or checkmate to your king
yellow king checkmated, but using this logic r/y win "first" - yellow take queen and red capture blue king
but what if instead blue bishop was green bishop: yellow capture queen ,green capture yellow king , red capture blue king and who was first? or the game just goes on green against red ?

> for me this questions is similar to [2pc examples]
The question is not similar: in 2pc king captures don't exist.
> in both black cannot checkmate white
this is falsy logic. checkmate just means the king will get captured on the next move. in any case it's slower than actually capturing the king. in both your 2pc examples, black will/would capture the white king BEFORE white can capture the black king (and that's precisely the reason why white is not allowed to make certain moves, they are equal to resignation, and of course white is allowed to resign if he wishes to do so.)

This example is not relevant to this topic. The suggested rule change is only for situations where a king can be captured, and thus only examples where a king capture is possible are meaningful to illustrate the rule.

This example shows why the suggested rule is not complete in a sense that the player who's king gets / would get captured first always loses. The capture the king variant is complete in this sense, and r/y wins if Green plays Qf1. But for the rule in question, nothing changes here as there is no king capture involved.
Same for the example with the pinned yellow rook: no king captures possible - nothing changes

Good example. The logic of the suggested rule is that Green is not checkmated here! Because Green has a way to get out of check: By capturing the red king, the red army dies, and Green is no longer in check. Taking the red K is as good as taking the red Q for eliminating the check. It's also much fairer imo, as the red king will get captured before the green K.
Something to note: Red's turn is immediately after Green's. By the logic of having checkmates only happen on a player's turn nothing changes. Assume it was Yellow checking Green, and Green captures the Yellow king in response, you could argue that the game should not end until Yellow's turn is reached, allowing for Red to potentially capture Blue's King, getting there first, and R/Y winning despite Yellow's king being taken before Blue's.
But this is not relevant, because it's impossible to capture the King of the player to your right! There is no possible way Green can capture the Yellow king, so this scenario doesn't need to be considered.

In the end, I feel this rule doesn't change the game much at all. King captures are rare. You just have to be a bit more careful your king and your teammate's king can't get captured, which I think makes a lot of sense. The current rules I feel have a small loophole allowing things that just don't look right or fair.
If green mates red, could yellow save red king by checking green?
I´d say, in teams yes, that´s a defensive resource (though useless if red king can not untangle out of the mating net, or yellow doesn´t have perpetual check/capture king chances).
Should Green be allowed to capture the Red King, thereby checkmating Red, ending the game?
(It's not possible under the current rules..)
Yes or No?
Similarly: