The rule book states that if you build a building on the same tile as a Great Person, you put the Great Person in your "reserves" and can place that Great Person in the outskirts of any of your cities at the start of your next turn.

What if you want to remove a Great Person from one city's outskirts and place them in another city's outskirts, but you are unable/unwilling to place a new building on the Great Person's tile? Can you just pick up or reassign Great People? If so, when do you do this?

Since Trade is calculated after Start of Turn and before City Management, and since you can replace the Great Person during City Management by building a building on that tile, and can place that Great Person at the Start of Turn of the next turn, there is no loss of effectiveness by the Great Person. So I don't see any reason why you shouldn't be able to just place Great People in your reserves during the City Management phase without replacing them with buildings.

What does everybody think about this?

As someone else pointed out in another post, the rules only tell you what you can do, not what you can't do. In other words, if the rules don't specifically state that you CAN do something, then you CAN'T.

If you do as you suggest, you could effectively shuffle around any and all Great People every single turn with no penalty. Simply place Great People at each Start of Turn phase, use them during the City Management phase, and then remove all of them from the board. Use them to crank out culture in your capital this turn, and then use them to boost production in some other city the next. You could also use this as a way to prevent loss of Great People from city losses or nuclear strike. That doesn't sound very fair.

I say don't do it, but if you must create a house rule allowing it, then restrict it considerably. First, remove them from the board during the Start of Turn phase, so you lose all of their benefit that entire turn. Secondly, only allow one such move per turn.

If the board game intent is to remain true to the video game then no you cannot move a great person once "settled" in a city. They can only move when stolen. They only way I see for the owner to move a great person is to destroy the great person my building on top.

I tend to agree. But the question did come up in one game that we were playing, and the rule book didn't answer the question. Most players were leaning towards following a strict interpretation of the book's rules: only allowing Great People to be moved when replaced with a building or wonder. But the fact that the game allows a Great Person to be put in the reserves during City Management (after they have already contributed to a city's production) and then placing them again during Start of Turn (allowing them to immediately contribute to their new city's production) set a precedent for allowing one of our players to argue that since the game designers did not apply a penalty to replacing the GP with a building, it would stand to reason that there should be no reason why GPs couldn't be moved at will during a player's turn. It was a fairly compelling argument, even though we ended up not allowing it.

I agree with the two responses so far that allowing a player to move GPs whilly-nilly would be overpowering and unfair. But I am curious now to know if anybody has come up with any interesting (and well-balanced) house rules that allow for players to move GPs without having to build a building to replace them. If anybody has any suggestions, I'd love to hear them!