Electrochemistry: Does ion flow from one half-cell to another?

My textbook says one of the functions of a salt bridge is that "it completes the circuit by allowing ions to move from one half-cell to another". However, according to my understanding, only the ions in the salt bridge move into the half-cells.  

[![enter image description here](https://i.stack.imgur.com/m165N.png)](https://i.stack.imgur.com/m165N.png)


Take the above chemical cell as an example, only the potassium ions and nitrate ions move into different half-cell respectively. The mobile ions in the two electrolyte do not move from one half-cell to another, right? 


