Jam_Man
Life President
The buying club will indeed pay the agent but in addition the agent will want his part of the players wages and fees,look at it this way if an agent has 100 players on his books and each player remained at one club during his playing career then the agent would earn nothing!
If the player stays at one club the agent earns from the wages, like you said about 20 words prior, plus each time a contract is renewed.
You are correct though it is often in the agents best interest for players to move as they get paid part of the signing on fee.
However they can still earn money if they dont, Wayne Rooney hasnt moved anywhere but his agent will still make a fortune from his salary and signing on fees every new contract.