That's exactly what happens. In effect, the lethargic are paying the active with the bank being the conduit for the flow of this money
Makes me wonder whether savers are influencing the rates though.
If loads of people flock to get the latest high rate, then that lender has loads of money to lend out, and thus they make a good profit. Along comes another lender, at a higher rate, and most people move their money again (they moved it to start, so will continue moving each time). Thus the first lender now has a shortfall of disposable cash to meet liabilities, and may have to increase their rates to get people to move their money back again. Or is this too simplistic?
The trick is going to be predicting peak, so that instead of easy access accounts, the money (or at least some of it) can be moved to a fixed rate one. There's little point right now with rates rising so fast, but that will eventually stop.