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AlphaZero has been used to more objectively study chess variants in this paper: https://arxiv.org/pdf/2009.04374.pdf

I don't think they mention anything about speeding up training. The original training time for AlphaZero was only a few hours anyway so I don't think that was ever a major constraint. I would imagine that each neural network performed best on the variant on which it was trained.

There is some discussion on reddit:

https://www.reddit.com/r/chess/comments/c4dgas/alpha_zero_fi...

If the piece movement rules are randomized, it's not going to be learning chess, is it...

Maybe they meant the piece type is randomized, because each piece has a different movement rule. This would result in things like maybe 5 pawns and 3 rooks instead of the normal numbers.

That would be Fischer Random Chess (aka Chess960). I know engines play it, but I don’t know if they’re trained on it.

That indeed randomizes the starting position (subject to some constraints), but it doesn’t randomize the abilities of the pieces.