It should not come as a surprise that elementary end-games can be completely different in Alice Chess. The pieces only superficially mimic the orthodox pieces, but in fact are completely different beasts: they are al locusts, which capture in passing, and should end on an empty square. So one cannot assume end-games like KBBK, KBNK or KNNK get the same result as in orthodox Chess. I am not sure if it has ever been studied whether these end-games are general wins.

KRK (and hence KQK) seems winnable, because a King is rather vulnerable in Alice Chess: it is not only forbidden to end on a square that is attacked, but also to pass through one. So Alice pieces are pretty good at confining a King; it does not really matter which board they are on. That means a Rook confines a King just like it would in orthodox Chess, and putting the Rook on the same rank or file as the bare King while you have opposition drives the bare King back in the same way, even if it would not actually check because it is on the other board. So the technique is pretty much the same.

Only for delivering the mate you have to be careful that the Rook actually checks, or it would be stalemate. Which is not entirely trivial, because pieces cannot triangulate in Alice Chess. But when a bare King gets confined to a1-b1 with the other King on b3, a Rook on the c-file can either go to c1 in one or two moves to get it on the right board, and to make sure the bare King is on a1 in both cases an extra King move b3-a3 can be played

The Troitsky study exploit the fact that two Knights can easily force stalemate, so that KNNK would be won if stalemate did not cause an instant draw (e.g. because it is an instant win, or forces the opponent to pass a turn). The extra Pawn provides moves that spoil the stalemate to achieve a similar effect, giving one of the Knights time to manoeuvre to the mating square while the black King stands trapped in a corner. But there still is a deadline in achieving this, as at some point the Pawn will promote, and the moves of the resulting Queen will no longer be irrelevant.

In the shown position the Knight is just in time to deliver mate before the Queen can be used, but in Alice Chess ends up on the wrong board for delivering check. But when Ne5 would have started on the other board, the line from the orthodox study would work just as well. Problem is that the Knight cannot triangulate, so you cannot get it to the same square in an odd number of moves to swap boards. So you should somehow force the black King to triangulate.

The problem in the Troitsky study is that the mate is really occurring in the wrong corner, far away from the Pawn, so that the other Knight (which must keep blocking the Pawn until the mate is only two moves away) cannot be used in the confinement, and you have very little control over what the black King does. So if there is a winning strategy here, it must be through driving the bare King to h1 wint King plus a single Knight, making use of the fact that f1, f3 and e4 form a no-go area for the bare King. In that corner you could use the 'free' Knight to guard h3, while moving Kg3-f2 to effectively lose a tempo:

I am not sure whether this will be possible. (And it also depends on whether the bare King arrived in the corner on a board where Nf3 would check it, or black would not be forced into the corner, but would advance his Pawn.) It would be interesting to generate End-Game Tables for Alice Chess. Since the Alice board has 2x64 cells, this would be harder than for orthodox Chess: each extra chess man gives another doubling of the size. For checkmating a bare King it doesn't matter on which board the other King is, though. So tables for end-games like KBNK are 8-times larger than for orthodox Chess.