Example moving knife cake cutting method
1) we assume that all players will accept that if they had as much opportunity to a given cake as every other player they are satisfied.
2) everyone will be fine with someone taking a small piece if that person is doing so under the impression that it is the same size
3) cakes are combinable...two small pieces can be pressed together to make one of the same size as them separately and everyone accepts that the new cake is the same as the old. This would work better if we were splitting a beer or wine.
4) People can cut perfectly and don't try to cheat when performing step 3. A referee might work for that. This is by far the weakest part of the method.
Step 1) A particpant is chosen at random to cut his piece of cake. He cuts off a piece he thinks is fair.
Step 2) Everyone is given the option to claim his piece is too big. The players who feel that way particpate in step 3. If no one feels that way, he can keep his piece, skip step 3, and be no longer part of the game.
Step 3) He holds his knife over the cake in an exact steady nature over the left hand side and slowly moves the knife to right above the cake. When each person feels he would cut it accurately if he cuts there they yell "STOP". He cuts when the last person says stop and that person gets the remaining right hand side of the cake. The player is no longer part of the game.
Step 4) The cake and all scrap is pressed together to make one big cake.
Step 5) Repeat all steps until there is only one player. He gets all remaining cake.
When N=2, this is equivalent to the standard answer you describe in your question as he will say "STOP" immediately.
If cakes cannot be pressed together, assume that any scrap will be given to the person to be awarded cake on the next round. If any players don't say stop until after the knife has passes over the piece of cake they are playing for, the knife will proceed over the scrap pile.
If you must allow for people to want pieces that are produced after they are given their piece the problem gets unsolvable for all N. For N=3, however, there are solutions. For example, if all players hold a knife over what they consider to be the center of the right hand side of a cake while a referee moves his knife from left to right, both cuts can effectively happen at the same time. When any player yells "CUT" the referee cuts and the yeller gets the cake to the left of the referee's knife. The other cut (on the right hand side of the cake) is made at the knife in the middle of the other 3 players. Besides the yeller the player with the leftmost knife gets the middle cake while the other player gets the rightmost cake.