No matter what method you decide on, you will need to keep track of two things, the current turn order and the next turn order. I have a few options you could try:
You can simply have two tracks that will show current and next turn orders. This is a simple solution, players can all stay in their seats and the turn will just jump around the table, this is good for certain games where the "player to your left" or "player to your right" kind of abilities do not exist.
Musical Chairs and Chips
This is good for games where "player to your left" or "player to your right" kind of abilities do exist. You have the players sit clockwise in turn order. And they will start each round holding a chip with their number on it. When switching turn orders throughout the game, players just switch these chips. At the start of each round, players will reseat themselves according to turn order.
Infinite Stacking Track
This is a bit complex, but the idea is that each player has a token with his color/team on it. And there is an "infinite" track, (this could just be a very long track) Initial turn order is decided and tokens are placed in that order on the track, whenever a player modifies a turn order, you simply move the token up or down the track, if a token already exists on that you place yours below that token.
This way, when a new turn is started, going from the first tile in the track towards the end and working from the top-most token down. You get an extra little mechanic where some tiles are blank, which means players after those tiles would need to work extra-hard to get ahead of those that are already passed them. The difference a regular track is that in a regular track you would just replace the tokens instead of stack them.
Double Secret Cards
This is a good method if at some point the turn order might be randomized, or you want to keep the turn order secret, similar to the Two Tracks suggestion you would have 2 sets of cards, one for the current turn, one for the next turn (maybe a different color on the back?) Players would start the game where both cards are the same number (turn order) and only when it's that players turn, he would reveal that he is the player with that number, (revealing the current card color) in this way, players can manipulate the turn order in secret (which adds an interesting element to the game).
The only pitfall is that players need to remember who they gave the next turn cards to, so they can also give them the current turn cards at the end of the turn.