Are there multiplayer equivalents to trading places after the first move?
Mathematically the Pie rule can be generalized. This is actually a well known problem.

One of the solutions is to let people take it in turns to add what they want from the common lot to a set of goods until one person calls that he wants that set. For example, in the 40 thieves problem, the goal is to share the bounty in 40 equal sets, which seems difficult, since there are jewels, gold, weapons etc. So the leader prepares a set and each thief adds one thing at a time until one thief claims he wants that set. Then the thief leaves with his share and the problem is reduced to n-1 people.

I never heard of a board game using exactly this principle, however in some board games the players bid point to determine the order of the players for the turn. For example, in the expert-variant of the game Tikal, each turn you draw as many tiles as there are players, and they bid to play first.

The strict application of the principle to a board game would be to have a first turn where you set up some initial position on the board. And afterwards, the players declare: "I add X money to the white meeple", "I add X pieces of wood to the white meeple", "I add X card to the blue meeple", "OK STOP, I take the blue meeple".

Actually, there are probably some games that use this kind of mechanic, I just don't recall one now.

[Edit:] You probably wonder what to do when several players wants to take a share. In that case, you just subtract something until nobody or (only one person) wants it any more, and then you continue adding.