The reason En Passant exists is to prevent the opponent from locking the structure by force when advancing a pawn deep into the opponent's territory. The rule makes attacking chess more effective and the game is more exciting and dynamic as a result.
From a game design perspective, I don't see how allowing other pieces to capture En Passant makes the game more enjoyable or interesting.
The En Passant move makes no sense unless it is available to all pieces. This would reinvigorate many endings but also some openings and middle games. If the pawn on e2 moves to e4 then any enemy piece attacking e3 should have the option to take that pawn en passant.
I do not actually consider this a variant. It should be the norm.
These captures can still only be performed on the immediate turn after the two-square move,.