I’m sorry for being late on this thread. On JA Magz template, we use the JA Multilingual component to build the multilingual site demo including the dropdown effect of the flags as you see.
In addition, I have checked your Magz site and saw that the multilingual process was not finished completely. Because you can see that the language menus were created in the same menu (Section Menu), instead of specific main menu as it is guided in our documentation and many other related tutorials.
In this case, I’d like to suggest you to try our JA Multilingual component and refer its documentation to get flags showing as you want.